Chapter # 12
1. Sol.
Simple Harmonic Motion
SOLVED EXAMPLES
The resultant force acting on a particle executing simple harmonic motion is 4 N when it is 5 cm away from the centre of oscillation. Find the spring constant. The simple harmonic motion is defined as F = – k x. The spring constant is k =
F x
4N 4N = 5 cm = = 80 N/m. 5 10 2 m
2.
A particle of mass 0.50 kg executes a simple harmonic motion under a force F = – (50 N/m)x. If it crosses the centre of oscillation with a speed of 10 m/s, find the amplitude of the motion.
Sol.
The kinetic energy of the particle when it is at the centre of oscillation is E=
1 mv2 2
1 (0.50 kg) (10 m/s)2 2 = 25 J. The potential energy is zero here. At the maximum displacement x = A, the speed is zero and hence the =
kinetic energy is zero. The potential energy here is
1 kA2 = 25 J 2 The force on the particle is given by F = – (50 N/m)x. Thus, the spring constant is k = 50 N/m. Equation (i) gives
or, 3. Sol.
1 2 kA . As there is no loss of energy,, 2 .............(i)
1 (50 N/m) A2 = 25 J 2 A = 1 m.
A particle of mass 200 g executes a simple harmonic motion. The restoring force is provided by a spring of spring cosntant 80 N/m. Find the time period. The time period is T = 2
= 2
m k 200 10 3 kg 80 N / m
= 2× 0.05 s = 0.31 s. 4.
A particle executes simple harmonic motion of amplitude A along the X-axis. At t = 0, the position of the particle is x = A/2 and it moves along the positive x - direction. Find the phase constant if the equation is written as x = A sin (t + ).
Sol.
We have x = A sin (t = ). At t = 0, x = A/2. Thus, A/2 = A sin or, sin = 1/2 or, = /6 or 5/6. The velocity is v =
dx = A cos(t + ). dt
manishkumarphysics.in
Page # 1
Chapter # 12 At
Simple Harmonic Motion t = 0, v = A cos.
5 3 3 = and cos =– 6 6 2 2 As v is positive at t = 0, must be equal to /6.
Now
cos
5.
A particle of mass 40 g execute a simple harmonic motion of amplitude 2.0 m. If the time period is 0.20 s, find the total mechanical energy of the system.
Sol.
The total mechanical energy of the system is 2
2 2 m A 2 2 1 = m A2 = 2 T T2
1 E= m 2 A2 2 =
6. Sol.
2 2 ( 40 10 3 kg)(2.0 10 2 m)2 (0.20 s)2
= 7.9 × 10–3 J
A body makes angular simple harmonic motion of amplitude /10 rad and time period0.05 s. If the body is at a displacement /10 rad at t = 0, write the equation giving the angular displacement as a function of time. Let the required equation be = 0 sin (t + ) Here
0 = amplitude = =
rad 10
2 2 = 0.05 s = 40 s–1 T
so that = rad sin 40 s 1 t 10 At t = 0, = /10 rad. Putting in (i)
..............(i)
= sin 10 10 or, sin =1 or, = /2 Thus by (i),
1 t = rad sin 40 s 2 10
7. Sol.
Calculate the time period of a simple pendulum of length one meter. The acceleration due to gravity at the place is 2 m/s2. The time period is T = 2 / g
8. Sol.
rad cos [(40 s–1) t ]. = 10
= 2
1.00 m 2m / s 2
= 2.0 s.
In a laboratory experiment with simple pendulum it was found that it took 36 s to complete 20 oscillations when the effective length was kept at 80 cm. Calculate the acceleration due to gravity from these data. The time period of a simple pendulum is given by T = 2 / g or,
g=
4 2
...........(i) T2 In the experiment described in the question, the time period is T=
36 s = 1.8 s. 20
thus, by (i) manishkumarphysics.in
Page # 2
Chapter # 12
Simple Harmonic Motion g=
4 2 0.80 (1.8 s)2
= 9.75 m/s2.
9.
A uniform rod of length 1.00 m is suspended through an end and is set into oscillation with small amplitude under gravity. Find the time period of oscillation.
Sol.
For small amplitude the angular motion is nearly simple harmonic and the time period is given by T = 2
= 2
mg = 2 3g = 2
(m 2 / 3) mg 1.00 m 3 9.80 m / s 2
= 1.16 s.
10.
A uniform disc of radius 5.0 cm and mass 200 g is fixed at its centre to a metal wire, the other end of which is fixed with a clamp. The hanging disc is rotated about the wire through an angle and is released. If the disc makes torsional oscillations with time period 0.20 s, find the torsional constant of the wire.
Sol.
The situation is shown in figure. The moment of inertia of the disc about the wire is
mr 2 (0.200 kg)(5.0 10 2 m)2 = 2 2 = 2.5 × 10–4 kg - m2. The time period is given by =
T = 2
or,
k=
42 T2
k =
4 2 (2.5 10 4 kg m 2 ) (0.20 s)
= 0.25
2
kg m 2 s2
.
11.
Find the amplitude of the simple harmonic motion obtained by combining the motions x1 = (2.0 cm) sint and x2 = (2.0 cm) sin (t + /3).
Sol.
The two equations given represent simple harmonic motions along X-axis with amplitudes A1 = 2.0 cm and A2 = 2.0 cm. The phase differnce between the two simple harmonic motions is /3. The resultant simple harmonic motion will have an amplitude A given by A= =
A 12 A 22 2A 1A 2 cos
( 2.0cm) 2 ( 2.0 cm)2 2( 2.0 cm) 2 cos
QUESTIONS
FOR
3
= 3.5 cm
SHORT
ANSWER
1.
A person goes to bed at sharp 10.00 pm every day. It is an example of periodic motion? If yes, what is the time period ? If no, why ?
2.
A particle executing simple harmonic motion comes to rest at the extreme positions. Is the resultant force on the particle zero at these positions according to Newton’s first law?
3.
Can simple harmonic motion take place in a noninertial frame? If yes, should the ratio of the force applied manishkumarphysics.in
Page # 3
Chapter # 12 Simple Harmonic Motion with the displacement be constant ? 4.
A particle executes simple harmonic motion. If you are told that its velocity at this instant is zero, can you say what is its displacement ? If you are told that its velocity at this instant is maximum, can you say what is its displacement ?
5.
A small creature moves with constant speed in a vertical circle on a bright day. Does its shadow formed by the sun on a horizontal plane move in a simple harmonic motion ?
6.
A particle executes simple harmonic motion. Let P be a point near the mean position and Q be a point near an extreme. The speed of the particle at P is larger then the speed at Q. Still the particle crosses P and Q equal number of times in a given time interval. Does it make you unhappy ?
7.
In measuring time period of a pendulum, it is advised to measure the time between consecutive passage through the mean position in the same direction. This is said to result in better accuracy than measuring time between consecutive passage through an extreme position. Explain.
8.
It is proposed to move a particle in simple harmonic motion on a rough horiozntal surface by applying an external force along the line of motion. Sketch the graph of the applied forceagainst the position of the particle. Note that the applied force has two values for a given position depending on whether the particle is moving in positive or negative direction.
9.
Can the potential energy in a simple harmonic motion be negative? Will it be so if we choose zero potential energy at some point other than the mean position ?
10.
The energy of a system in simple harmonic motion is given by E =
1 m2A2. Which of the following two 2
statements is more appropriate ? (A) The energy is increased because the amplitude is increased. (B) The amplitude is increased because the energy is increased. 11.
A pendulum clock gives correct time at the equator. Will it gain time or loose time as it is taken to the poles ?
12.
Can a pendulum clock be used in an earth satellite ?
13.
A hollow sphere filled with water is used as the bob of a pendulum. Assume that the equation for simple pendulum is valid with the distance between the point of suspension and centre of mass of the bob acting as the effective length of the pendulum. If water slowly leaks out of the bob, how will the time period vary ?
14.
A block of known mass is suspended from a fixed support through a light spring. Can you find the time period of vertial oscillation only by measuring the extension of the spring when the block is in equilibrium.
15.
A platoon of soldiers marches on a road in steps according to the sound of a marching band. The band is stopped and the soldiers are ordered to break the steps while crossing a bridge. Why ?
16.
The force acting on a particle moving along X-axis is F = – k(x – v0t) where k is a positive cosntant. An observer moving at a constant velocity v0 along the X-axis looks at the particle. What kind of motion does he find for the particle?
Objective - I 1.
A student says that he had applied a force F – k/x on a particle and the particle moved in simple harmonic motion. He refuses to tell whether k is a constant or not. Assume that he has worked only with positive x and no other force acted on the particle. ,d fo|kFkhZ dgrk gS fd mlus fdlh d.k ij cy F – k/x yxk;k gS rFkk d.k ljy vkorZ xfr dj jgk gSA og ;g crkus ls euk dj jgk gS fd k fu;r gS ;k ugha A ;fn ;g eku fy;k tk;s fd mldk dk;Z dsoy /kukRed x fn'kk
ds fy;s gh gS rFkk d.k ij vU; dksbZ cy dk;Zjr ugha gS & (A*) As x increases k increases x c<+u s ds lkFk k c<+rk gSA (B) As x increases k decreases x c<+u s ds lkFk k de gksrk gSA manishkumarphysics.in
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Chapter # 12 (C) (D) 2.
Simple Harmonic Motion As x increases k remains constant. x c<+u s ds The motion cannot be simple harmonic. xfr ljy
lkFk k fu;r jgrk gSA vkorhZ ugha gks ldrh gSA
The time period of a particle in simple harmonic motion is equal to the time between consecutive appearance of the particle at a particular point in its motion. This point is
ljy vkorZ xfr djrs gq, d.k ds fdlh fof'k"V fcUnq ij Øekxr :i ls n`f"Vxkspj gksus ds le;kUrjky dks d.k dk vkorZdky dgrs gSA ;g fcUnq gS & (A) the mean position ek/; fLFkfr (B*) an extreme position vf/kdre foLFkkiu dh fLFkfr (C) between the mean position and the positive extreme
/kukRed vf/kdre foLFkkiu rFkk ek/; fLFkfr ds e/; (D) between the mean position and the negative extreme.
_.kkRed vf/kdre foLFkkiu rFkk ek/; fLFkfr ds e/; 3.
The time period of a particle in simple harmonic motion is equal to the smallest time between the particle acquiring a particular velocity v . The value of v is ljy vkorhZ xfr esa d.k dk vkorZdky ml U;wure le; ds cjkcj gksrk gS] ftlesa d.k ,d fof'k"V osx v izkIr djrk gSA v dk eku gS & (A*) vmax (B) 0 (C) between 0 and vmax 0 o vmax ds chp (D) between 0 and –vmax 0 o –vmax ds chp
4.
The displacement of a particle in simple harmonic motion in one time period is
ljy vkorhZ xfr esa ,d vkorZdky esa d.k dk foLFkkiu gksrk gS & (A) A 5.
(B) 2A
(C) 4A
(A) A
(B) 2A
(C*) 4A
(A) A 2
(B) A 2 /2
9.
(C) A 2 / 2
(D*) zero. 'kwU;
The motion of a particle is given by x = A sin t + B cos t. The motion of the particle is fdlh d.k dh xfr x = A sin t + B cos t }kjk O;ä dh tkrh gSA d.k dh xfr gS & (A) not simple harmonic (B) simple harmonic with amplitude A + B (C) simple harmonic with amplitude (A + B)/2
(D*) simple harmonic with amplitude
(A) ljy
(B) ljy
vkorhZ ugha gSA (C) ljy vkorhZ ftldk vk;ke (A + B)/2 gS 8.
(D) zero 'kwU;
The average acceleration in one time period in a simple harmonic motion is
ljy ^vkorhZ xfr* esa ,d vkorZdky esa vkSlr Roj.k gksrk gS & 7.
'kwU;
The distance moved by a particle in simple harmonic motion in one time period is
ljy vkorhZ xfr esa ,d vkorZdky esa d.k ds }kjk r; dh xbZ nwjh gksrh gS & 6.
(D*) zero
A 2 B2
vkorhZ ftldk vk;ke A + B gSA (D*) ljy vkorhZ ftldk vk;ke A 2 B 2 gS
The displacement of a particle is given by r = A ( i cost + j sint). The motion of the particle is fdlh d.k dk foLFkkiu r = A ( i cost + j sint) }kjk O;ä fd;k tkrk gSA d.k dh xfr gS & (A) simple harmonic (B) on a straight line (C*) on a circle (D) with constant acceleration (A) ljy vkorhZ (B) ljy js[kk ds vuqfn'k (C*) o`Ùkh; iFk ij (D) fu;r Roj.k ds lkFk A particle moves on the X-axis according to the equation x = A + B sin t. The motion is simple harmonic with amplitude ,d d.k X-v{k ds vuqfn'k lehdj.k x = A + B sin t ds vuqlkj xfr'khy gSA xfr ljy vkorhZ gS] ftldk vk;ke
gS &
(A) A 10.
(B*) B
(C) A + B
(D)
A 2 B2
Figure represents two simple harmonic motions. The parameter which has different values in the two motions is HCV_Ch-12_Obj I_10 manishkumarphysics.in
Page # 5
Chapter # 12
Simple Harmonic Motion
fp=k esa nks ljy vkorZ xfr;k¡ fn[kk;h x;h gSA nksuksa xfr;ksa esa fdu izkpyksa ds vyx&vyx eku gksaxsA
(A) amplitude (A) vk;ke 11.
(B) frequency (B) vko`fÙk
(C*) phase (C*) dyk
(D) maximum velocity (D) vf/kdre osx
The total mechanical energy of a spring - mass system in simple harmonic motion is
1 m 2A2. Suppose the oscillating particle is replaced by another particle of double the mass while the 2 amplitude A remains the same. The new mechanical energy will E=
1 2
ljy vkorhZ xfr esa fLizax nzO;eku fudk; dh dqy ;kaf=kd ÅtkZ E = m 2A2 gSA eku fyft;s fd nksyu djus okys d.k dks nqxus nzO;eku okys d.k ls izfrLFkkfir dj fn;k tkrk gS tcfd vk;ke A vifjofrZr j[kk tkrk gSA u;h ;kaf=kd ÅtkZ & (A) become 2 E (A) 2 E gks tk;sxh 12.
(B) become E/2 (B) E/2 gks tk;sxh
(C) become (C) 2 E gks
2E
tk;sxh
(D*) remain E (D*) E gh jgsxhA
The average energy in one time period in simple harmonic motion is
ljy vkorhZ xfr esa ,d vkorZdky esa vkSlr ÅtkZ gksrh gS & (A*) 13.
1 m 2 A2 2
1 m 2 A2 4
(D) zero 'kwU;
(C) m 2 A2
A particle executes simple harmonic motion with a frequency v. The frequency with which the kinetic energy oscillates is ,d d.k v vko`fr ds lkFk ljy vkorhZ xfr dj jgk gSA xfrt ÅtkZ ftl vko`fr ls nksyu dj jgh gS] og gS& (A) v/2
14.
(B)
(B) v
(D) zero 'kwU;
(C*) 2v
A particle executes simple harmonic motion under the restoring force provided by a spring. The time period is T. If the spring is divided in two equal parts and one part is used to continue the simple harmonic motion, the time period will (A) remain T (B) becomes 2T (C) become T/2 (D*) become T/ 2 ,d fLizax }kjk yxk;s x;s izR;ku;u cy ds izHkko esa ,d d.k ljy vkorhZ xfr dj jgk gSa vkorZdky T gSA ;fn fLizax
dks nks cjkcj Hkkxksa esa foHkkftr djds] ,d Hkkx dk mi;ksx ljy vkorhZ xfr ds fy;s fd;k tkrk gS] rks vkorZdky& (A) T jgsxk (B) 2T gks tk;sxk (C) T/2 gks tk;sxk (D*) T/ 2 gks tk;sxk 15.
Two bodies A and B of equal mass are suspended from two separate massless springs of spring cosntant k1 and k2 respectively. If the bodies socillate vertically such that their maximum velocities are equal, the ratio of the amplitude of A to that of B is HCV_Ch-12_Obj I_15 leku nzO;eku ds nks fi.M A o B nks vyx&vyx nzO;eku jfgr fLizaxksa ftuds fLizax fu;rkad k1 o k2 gS ls yVdk;s x;s gSA ;fn nksuksa fi.M Å/okZ/kj bl izdkj nksyu djrs gS fd mues vf/kdre osx leku gS A o B ds vk;keksa esa vuqikr
gksxkA
(A) k1 /k2 16.
(B)
k1 / k 2
(C) k2/k1
(D*)
k 2 / k1
A spring mass system oscillates with a frequency v. If it is taken in an elevator slowly accelerating upward, the frequency will (A) increase (B) decrease (C*) remain same (D) become zero ,d fLizax nzO;eku fudk; v vko`fr ls nksyu dj jgk gSA ;fn bldks Åij dh vksj /khes Roj.k ls xfr'khy fy¶V esa
ys tk;k tk;s rks bldh vko`fr& (A) c<+ tk;sxhA (B) de gks tk;sxhA
(C*) vifjofrZr
manishkumarphysics.in
jgsxh
(D) 'kwU;
gks tk;sxhA Page # 6
Chapter # 12 17.
Simple Harmonic Motion
A spring-mass system oscillates in a car. If the car accelerates on a horizontal road, the frequency of oscillation will (A) increase (B) decrease (C*) remain same (D) become zero.
,d fLizax nzO;eku fudk; ,d dkj esa nksyu dj jgk gSA ;fn dkj {ksfrt lM+d ij Rofjr gksus yx tkrh gS] rks vkorZdky (A) c<+ tk;sxk (B) de gks tk;sxkA (C*) vifjofrZr jgsxkA (D) 'kwU; gks tk;sxkA 18.
A pendulum clock that keeps correct time on the earth is taken to the moon. It will run (A) at correct rate
(B) 6 times faster
(C)
6 times faster
(D*) 6 times slower
i`Foh ij lgh le; n'kkZus okyh yksyd ?kM+h pUnzek ij ys tk;h tkrh gSA ;g pysxh& (A) leku nj ls (B) 6 xuh rst (C) 6 xqu h rst (D*) 6 xuh /kheh 19.
A wall clock uses a vertical spring-mass system to measure the time. Each time the mass reaches an extreme position, the clock advances by a second. The clock gives correct time at the equator. If the clock is taken to the poles it will (A) run slow (B) run fast (C) stop working (D*) give correct time.
,d nzO;eku ?kM+h esa le; ekiu gsrq m/okZ/kj fLizax nzO;eku fudk; dk mi;ksx fd;k tkrk gSA izR;sd ckj nzO;eku tc fLFkfr rd igqaprk gSA ?kM+h 1 lsd.M vkxs c<+rh gSA ?kM+h fo"kqor~ js[kk ij lgh le; n'kkZrh gS] ;fn bl ?kM+h dks /kzqoksa ij ys tk;k tk;s rks ;g& (A) /kheh pysxh A (B) rst pysxhA (C) dk;Z djuk cUn dj nsxhA (D*) lgh le; n'kkZxhA 20.
A pendulum clock keeping correct time is taken to high altitudes, (A) it will keep correct time (B) its length should be increased to keep correct time (C*) its length should be decreased to keep correct time (D) it cannot keep correct time even if the length is changed.
lgh le; n'kkZus okyh yksyd ?kM+h Å¡ps LFkku ij ys tkus ij& (A) lgh le; n'kkZ;xhA (B) lgh le; n'kkZu s ds fy, bldh yEckbZ c<+kuh iM+sxhA (C*) lgh le; n'kkZu s ds fy, bldh yEckbZ de djuh gksxhA (D) yEckbZ esa ifjorZu djus ij Hkh ;g lgh le; ugh n'kkZ;sxhA 21.
The free end of a simple pendulum is attached to the ceiling of a box. The box is taken to a height and the pendulum is oscillated. When the bob is at its lowest point, the box is released to fall freely. As seen from the box during this period, the bob will (A) continue its oscillation as before (B) stop (C*) go in a circular path (D) move on a straight line.
ljy yksyd dk eqDr fljk ,d ckWDl dh Nr ls tqM+k gqvk gSA ckWDl dks fdlh Å¡pkbZ ij ys tk;k tkrk gS rFkk yksyd nksyu dj jgh gSA tc ckWc fuEure fcUnq ij gS] ckWDl dks NksM+ fn;k tkrk gSA ftlls ;g eqqDr :i ls fxjus yxrk gSA ckWDl esa ls ns[kus ij bl dky esa ckWc& (A) iwoZ dh Hkkafr gh nksyu djrk gSA (B) :d tkrk gSA (C*) o`Ùkkdkj iFk ij xfr djrk gSA (D) ljy js[kk ds vuqfn'k xfr djrk gSA
Objective - II 1.
Select the correct statements. (A*) a simple harmonic motion is necessarily periodic (B*) a simple harmonic motion is necessarily oscillatory (C) an oscillatory motion is necessarily periodic (D) a periodic motion is necessarily oscillatory
lgh dFku pqfu;sA (A*) ljy vkorhZ xfr fuf'pr :i ls vkorhZ xfr gksrh gSA (B*) ljy vkorhZ xfr fuf'pr :i ls nksy uh xfr gksrh gSA (C) nksy uh xfr fuf'pr :i ls vkoZfr xfr gksrh gSA (D) vkoZrh xfr fuf'pr :i ls nksyu xfr gksrh gSA manishkumarphysics.in
Page # 7
Chapter # 12 Simple Harmonic Motion 2. A particle moves in a circular path with a uniform speed. Its motion is (A*) periodic (B) oscillatory (C) simple harmonic (D) angular simple harmonic
o`Ùkkdkj iFk ij ,d d.k fu;r pky ls xfr'khy gSA bldh xfr gS& (A*) vkorhZ (B) nks y uh (C) ljy vkorhZ (D) dks.kh; ljy vkorhZ 3.
A particle is fastened at the end of a string and is whirled in a vertical circle with the other end of the string being fixed. The motion of the particle is (A*) periodic (B) oscillatory (C) simple harmonic (D) angular simple harmonic
,d Mkjh ds fljs ij ,d d.k fpidk;k x;k gS rFkk Mkjh ds nwljs fljs dks fLFkj j[krs gq, bls m/okZ/kj o`Ùk esa ?kqek;k tkrk gSA d.k dh xfr gSA (A*) vkorhZ (B) nks y uh (C) ljy vkorhZ (D) dks.kh; ljy vkorhZ 4.
A particle moves in a circular path with a continuously increasing speed. Its motion is (A) periodic (B) oscillatory (C) simple harmonic (D*) none of them
o`Ùkkdkj iFk ,d d.k fujUrj c
6.
The motion of a torsional pendulum is (A*) periodic (C) simple harmonic
(B*) oscillatory (D*) angular simple harmonic
ejksM+h nksyd dh xfr gksrh gSA (A*) vkorhZ (C) ljy vkorhZ
(B*) nksy uh (D*) dks.kh;
Which are always negative in a simple harmonic motion ? of the following quantities (A) F . a (B) v . r (C*) a . r (D*) F . r
(A) F . a
(B) v . r
(C*) a . r
(A*) F . a
(B) v . r
(C) a . r
(D) F . r
Which are always zero in a simple harmonic motion ? of the following quantities (A*) F × a (B*) v × r (C*) a × r (D*) F × r
fdlh ljy vkorZ xfr esa fuEu esa ls dkSulh jkf'k;k¡ lnSo 'kwU; gksrh gS& (A*) F × a 9.
(D*) F . r
Which are always positive in a simple harmonic motion ? of the following quantities (A*) F . a (B) v . r (C) a . r (D) F . r
fdlh ljy vkorhZ xfr esa fuEu esa ls dkSulh jkf'k;k¡ lnSo /kukRed gksrh gS& 8.
ljy vkorhZ
ljy vkorhZ
fdlh ljy vkorhZ xfr esa fuEu esa ls dkSulh jkf'k;k¡ lnSo _.kkRed gksrh gS& 7.
(D*) dks.kh;
(B*) v × r
(C*) a × r
(D*) F × r
Suppose a tunnel is dug along a diameter of the earth. A particle is dropped from a point, a distance h directly above the tunnel. The motion of the particle as seen from the earth is (A) simple harmonic (B) parabolic (C*) on a straight line (D*) periodic eku yhft;s fd i`Foh ds O;kl ds vuqfn'k ,d lqjax [kksnh x;h gSA lqjax ds Bhd Åij h Å¡pkbZ ij fLFkr fcUnq ls]
,d d.k fxjkrk tkrk gSA i`Foh ls ns[kus ij d.k dh xfr fn[kkbZ nsxh& (A) ljy vkorhZ (B) ifjoyf;d (C*) ljy js[kk ds vuqfn'k (D*) vkorhZ 10.
For a particle executing simple harmonic motion, the acceleration is proportional to (A*) displacement from the mean position (B) distance from the mean position (C) distance travelled since t = 0 (D) speed
ljy vkorZ xfr dj jgs d.k ds fy;s] Roj.k lekuqikrh gksrk gS& (A*) ek/; fLFkfr ls foLFkkiu ds (B) ek/; fLFkfr ls nwjh ds manishkumarphysics.in
Page # 8
Chapter # 12 (C) t = 0 ls (D) pky 11.
Simple Harmonic Motion
r; dh xbZ nwjh ds
A particle moves in the X-Y plane according to the equation r = ( i + 2 j ) A cost. The motion of the particle is (A*) on a straight line (B) on an ellipse (C*) periodic fuEu lehdj.k ds vuqlkj ,d d.k X-Y ry esa ljy vkorZ xfr dj r = ( i + 2 j ) A cost.
d.k dh xfr gS& (A*) ljy js[kk ds vuqfn'k (B) nh/kZo`Ùk ds vuqfn'k Sol.
r = ( ˆi + 2 ˆj ) A cos t
(D*) simple harmonic
jgk gSA
(C*) vkorhZ
(D*) ljy
vkorhZ
This is combination of S.H.M. along x and y axis having equations x = A cos t & y = 2A cos t This is equation of S.H.M. along a sraight line y = 2x. S.H.M. is always periodic. r = ( ˆi + 2 ˆj ) A cos t
;g x vkSj y v{k ds vuqfn'k ljy vkorZ xfr;ksa dk la;kstu gS ftuds lehdj.k x = A cos t vkSj y = 2A cos t ;g ljy js[kk y = 2x ds vuqfn'k ljy vkorZ xfr dk lehdj.k gSA l0 vk0 x0 lnSo vkorhZ gksrk gSAa 12.
A particle moves on the X-axis according to the equation x = x0 sin2 t. The motion is simple harnomic (A) with amplitude x0 (B) with amplitude 2x0 2 (C) with time period (D*)with time period ,d d.k lehdj.k x = x0 sin2 ds vuqlkj x-v{k ds vuqfn'k xfr'khy gSA xfr ljy vkorhZ gS& (A) x0 vk;ke ds lkFk (B) 2x0 vk;ke ds lkFk 2 (C) vkorZdky ds lkFk (D*vkorZdky ds lkFk
13.
In a simple harmonic motion (A) the potential energy is always equal to the kinetic energy (B) the potential energy is never equal to the kinetic energy (C) the average potential energy in any time interval is equal to the average kinetic energy in that time interval (D*) the average potential energy in one time period is equal to the average kinetic energy in this period.
ljy vkorZ xfr esa& (A) fLFkfrt ÅtkZ lnSo xfrt ÅtkZ ds cjkcj jgrh gS (B) fLFkfrt ÅtkZ lnSo xfrt ÅtkZ ls (C) fdlh le;kajky esa vkSlr fLFkfrt ÅtkZ ml le;karjky esa vkSlr xfrt ÅtkZ ds cjkcj jgrh gSA (D*) ,d vkorZkdy esa vkSlr fLFkfrt ÅtkZ bl vkorZdky esa vkSlr xfrt ÅtkZ ds cjkcj gksrh gSA 14.
In a simple harmonic motion (A*) the maximum potential energy equals the maximum kinetic energy (B*) the minimum potential energy equals the minimum kinetic energy (C) the minimum potential energy equals the maximum kinetic energy (D) the maximum potential energy equals the minimum kinetic energy
ljy vkorZ xfr esa& (A*) vf/kdre fLFkfrt ÅtkZ] vf/kdre xfrt ÅtkZ ds cjkcj gksrh gSA (B*) U;wu re fLFkfrt ÅtkZ] U;wu re xfrt ÅtkZ ds cjkcj gksrh gSA (C) U;wu re fLFkfrt ÅtkZ] vf/kdre xfrt ÅtkZ ds cjkcj gksrh gSA (D) vf/kdre fLFkfrt ÅtkZ] U;wure xfrt ÅtkZ ds cjkcj gksrh gSA 15.
An object is released from rest. The time is takes to fall through a distance h and the speed of the object as it falls through this distance are measured with a pendulum clock. The entire apparatus is taken on the moon manishkumarphysics.in
Page # 9
Chapter # 12 Simple Harmonic Motion and the experiment is repeated (A*) the measured times are same (B*) the measured speeds are same (C) the actual times in the fall are equal (D) the actual speeds are equal ,d oLrq dks fojkeoLFkk ls fxjk;k tkrk gSA h nwjh rd fxjus esa yxk le; rFkk blh nwjh rd fxjus
ij pky dk ekiu yksyd ?kM+h }kjk fd;k tkrk gSA ;fn lEiw.kZ midj.k dks pkan ij ys tk;k tk;s rFkk bl iz;ksx dks nksgjk;k tk;s& (A*) ekis x;s le; leku gksxsa (B*) ekih x;h pkysa leku gksxhaA (C) fxjus esa yxs okLrfod le; leku gksxsaA (D) okLrfod pkysa ,d leku gksxhaA
16.
Which of the following will change the time period as they are taken to moon ? (A*) A simple pendulum (B*) A physical pendulum (C) a torsional pendulum (D) a spring mass system
pUnzek ij ys tkus ij fuEu ls fdlds vkorZdky ifjofrZr gks tk;sxas& (A*) ljy yksy d (B*) HkkSfrd yksyd (C) ejksM+h nksy d (D) fLizax nzO;eku fudk;
WORKED OUT EXAMPLES 1. Sol.
1 The equation of particle executing simple harmonic motion is x = (5 m) sin ( s )t amplitude, time period and maximum speed. Also find the velocity at t = 1 s. Comparing with equation x = A sin (t + ), we see that the amplitude = 5 m,
and time period =
. Write down the 3
2 2 = = 2s. s 1
The maximum speed = A = 5 m × s–1 = 5 m/s. The velocity at time t = At
dx = A cos (t + ) dt
t = 1 s, 5 v = (5 m) ( s–1) cos = – m/s. 5 2
2.
A block of mass 5 kg executes simple harmonic motion under the restoring force of a spring. The amplitude and the time period of the motion are 0.1 m and 3.14 s respectively. Find the maximum force exerted by the spring on the block.
Sol.
The maximum force exerted on the block is kA when the block is at the extreme position.
2 = 2 s–1 T = k = m2 = (5 kg) (4 s–2) = 20 N/m. = kA = (20 N/m) (0.1 m) = 2 N.
The angular frequency = The spring constant Maximum force 3.
A particle executing simple harmonic motion has angular frequency 6.28 s–1 and ampitude 10 cm. Find (a) the time period, (b) the maximum speed, (c) the maximum acceleration, (d) the speed when the displacement is 6 cm from the mean position, (e) the speed at t = 1/6 s assuming that the motion starts from rest at t = 0.
Sol.
(a) (b) (c)
2 2 = s = 1 s. 6.28 Maximum speed = A = (0.1 m) (6.28 s–1) = 0.628 m/s. Maximum acceleration = A2 = (0.1 m) (6.28 s–1)2 = 4 m/s2.
Time period =
manishkumarphysics.in
Page # 10
Chapter # 12
Simple Harmonic Motion
(d)
v=
At
t=
–1 2 2 A 2 x 2 = (6.28 s ) (10 cm ) ( 6 cm ) = 50.2 cm/s. (e) At t = 0, the velocity is zero i.e., the particle is at an extreme. The equation for displacement may be written as x = A cost. The velocity is v = – A sin t.
1 s, 6
6.28 v = – (0.1 m) (6.28 s–1) sin 6
= ( – 0.628 m/s) sin 4. Sol.
3
= 54.4 cm/s.
A particle executes a simple harmonic motion of time period T. Find the time taken by the particle to go directly from its mean position to half the amplitude. Let the equation of motion be x = A sin t. At t = 0, x = 0 and hence the particle is at its mean position. Its velocity is v = A cos t = A which is positive. So it is going towards x = A/2. The particle will be at x = A/2, at a time t where
A = A sin t 2 or, sin t = 1/2 or, t = /6 Here minimum positive value of t is chosen because we are interested in finding the time taken by the particle to directly go from x = 0 to x = A/2. Thus, t = 5. Sol.
T = = . 6(2 / T ) 6 12
A block of mass m hangs from a vertical spring of spring constant k. If it is displaced from its equilibrium position, find the time period of oscillations. Suppose the length of the spring is stretched by a length . The tension in the spring is k and this is the force by the spring on the block. The other force on the block is mg due to gravity. For equilibrium, mg = k or = mg/k. Take this position of the block as x = 0. If the block is further displaced by x, the resultant mg x – mg = kx. force is k k
Thus, the resultant force is proportional to the displacement. The motion is simple harmonic with a time
m . k We see that in vertical oscillations, gravity has no effect on time period. The only effect it has is to shift the equilibrium position by a distance mg/k as if the natural length is increased (or decreased if the lower end of the spring is fixed) by mg/k. period T = 2
6.
A particle suspended from a vertical spring oscillates 10 times per second. At the highest point of oscillation the spring becomes unstretched. (a) Find the maximum speed of the block. (b) Find the speed when the spring is stretched by 0.20 cm. Take g = 2 m/s2.
Sol.(a) The mean position of the particle during vertical oscillations is mg/k distance away from its position when the spring is unstretched. at the highest point, i.e., at an extreme position, the spring is unstretched.
manishkumarphysics.in
Page # 11
Chapter # 12
Simple Harmonic Motion
Hence the amplitude is
mg . k The angular frequency is A=
........(i)
k = 2 v = (20) s–1 m
=
........(ii)
1 m = s 2. k 400 2 Putting in (i), the amplitude is or,
(b)
1 m 1 s 2 2 A = = m = 0.25 cm. 2 2 400 400 s The maximum speed = A = (0.25 cm) (20 s–1) = 5 cm/s. When the spring is stretched by 0.20 cm, the block is 0.25 cm – 0.20 cm = 0.05 cm above the mean position. The speed at this position will be
v=
A 2 x2
= (20 s–1)
(0.25 cm)2 (0.05 cm)2
15.4 cm/s. 7.
The pulley shown in figure has a moment of inertia about its axis and mass m. Find the time period of vertical oscillation of its centre of mass. The spring has spring constant k and the string does not slip over the pulley.
Sol.
Let us first find the equilibrium position. For rotational equilibrium of the pulley, the tensions in the two strings should be equal. Only then the torque on the pulley will be zero. Let this tension be T. The extension of the spring will be y = T/k, as the tension in the spring will be the same as the tension in the string. For translational equilibrium of the pulley, 2T = mg
or,
2 ky = mg
mg y = 2k
or,
mg the spring is extended by a distance 2 k when the pulley is in equilibrium. Now suppose, the centre of the pulley goes down further by a distance x. The total increase in the length of the string plus the spring 2x (x on the left of the pulley and x on the right). As the string has a constant length, the extension of the spring is 2x. The energy of the system is U=
mg 1 1 1 2 x 2 + mv2 – mgx + k 2 2 2 2k
2
manishkumarphysics.in
Page # 12
Chapter # 12
Simple Harmonic Motion m 2 g2 m v2 + + 2 kx2. 8k r2
1 = 2
As the system is conservative,
dU = 0, giving, dt
dv 0 = 2 m v + 4 kxv dt r
or,
or,
dv 4k x =– dt 2 m r
4k
a = – 2x where 2 =
2 m r Thus, the centre of mass of the pulley executes a simple harmonic motion with time period. T = 2
m /( 4 k ) . 2 r
8.
The friction coefficient between the two blocks shown in figure is µ and the horizontal plane is smooth. (a) If the system is slightly displaced and released, find the time period. (b) Find the magnitude of the frictional force between the blocks when the displacement from the meanposition is x. (c) What can be the maximum amplitude if the upper block does not slip relative to the lower block ? fp=k esa n'kkZ;s x;s nks xqVdksa ds e/; ?k"kZ.k xq.kkad µ gS ,oa {kSfrt ry fpduk gSA (a) ;fn fudk; FkksM+k lk foLFkkfir djds NksM+ fn;k tkrk gS rks vkorZdky Kkr djksA (b) tc ek/; fLFkfr ls foLFkkiu x gS rks xqVdksa ds e/; ?k"kZ.k cy dk ifjek.k Kkr djksA (c) ;fn fupys xqVds ds lkis{k Åijh xqVdk ugha fQlyrk gS rks vf/kdre vk;ke fdruk gks ldrk gS \ [M.Bank(07-08)_HCV__Ch.12_WOE_Q.8]
Sol.
(a)
For small amplitude, the two blocks oscillate together. The angular frequency is =
k Mm
Mm . k The acceleration of the blocks at displacement x from the mean position is and so the time period T = 2
(b)
kx a = – 2x = Mm The resultant force on the upper block is, therefore, mkx ma = Mm This force is provided by the friction of the lower block. Hence, the magnitude of the frictional force is mk | x | Mm
(c)
mk A at the Mm extreme positions. But the maximum frictional force can only be µ mg. Hence
Maximum force of friction required for simple harmonic motion of the upper block is
mk A = µ mg Mm 9.
or,
A=
µ(M m)g k
The left block in figure collides inelastically with the right block and sticks to it. Find the amplitude of the resulting simple harmonic motion. manishkumarphysics.in
Page # 13
Chapter # 12
Sol.
Simple Harmonic Motion
Assuming the collision to last for a small interval only, we can apply the principle of conservation of momentum. 2
v v 1 1 The common velocity after the collision is . The kinetic energy = (2m) = mv2. This is also the 2 2 2 4
total energy of vibration as the spring is unstretched at this moment. If the amplitude is A, the total energy can also be written as
1 kA2. Thus, 2
1 2 1 kA = mv2, giving A = 2 4
m 2 k v..
10.
Describe the motion of the mass m shown in figure. The walls and the block are elastic.
Sol.
The block reaches the spring with a speed v. It now compresses the spring. The block is decelerated due to
1 1 mv2 = kx2 and return back. It is accelerated due to the spring force 2 2 till the spring acquires its natural length. The contact of the block with the spring is now broken. At this instant it has regained its speed v (towards left) as the spring is unstretched and no potential energy is the spring force, comes to rest when
sotred. This process takes half the period of oscillation i.e., m / k . The block strikes the left wall after a time L/v and as the collision is elastic, it rebounds with the same speed v. After a time L/v, it again reaches the spring and the process is repeated. The block thus undergoes periodic motion with time period m / k 2L + . v 11.
Sol.
A block of mass m is suspended from the ceiling of a stationary elevator through a spring of spring constant k. Suddenly, the cable breaks and the elevator starts falling freely. Show that block now executes a simple harmonic motion of amplitude mg/k in the elevator. When the elevator is stationary, the spring is stretched to support the block. If the extension is x, the tension is kx which should balance the weight of the block.
Thus, x = mg/k. As the cable breaks, the elevator starts falling with acceleration ‘g’. We shall work in he frame of reference of the elevator. Then we have to use a psuedo force mg upward on the block. This force will ‘balance’ the weight. Thus, the block is subjected to a net force kx by the spring when it is at a distance x from the position of unstretched spring. Hence, its motion in the elevator is simple harmonic with its mean position corresponding to the unstretched spring. Initially, the spring is stretched by x = mg/k, where the velocity of the block (with respect to the elevator) is zero. Thus, the amplitude of the resulting simple harmonic motion is mg/k. 12.
The spring shown in figure is kept in a stretched position with extension x0 when the system is released. Assuming the horizontal surface to be frictionless, find the frequency of oscillation.
Sol.
Considering “the two blocks plus the spring” as a system there is no external resultant force on the system. Hence the centre of mass of the system will remain at rest. The mean positions of the two simple harmonic motions occur when the spring becomes unstretched. If the mass m moves towards right through a distance manishkumarphysics.in Page # 14
Chapter # 12 Simple Harmonic Motion x and the mass M moves towards left through a distance X before the spring acquires natural length, x + X = x 0. .............(i) x and X will be the amplitudes of the two blocks m and M respectively. As the centre of mass should not change during the motion, we should also have mx = MX. .............(ii) Mx 0 mx 0 From (i) and (ii), x = and X = . Mm Mm Mx 0 Hence, the left block is x = distance away from its mean position in the beginning of the motion. The Mm force by the spring on thick block at this isntant is equal to the tension of spring i.e., T = kx0. Now
x=
Mx 0 or,, Mm
Thus,
T=
k(M m) x or, M
x0 =
Mm x M a=
T k(M m) = x. m Mm
The angular frequency is, therefore, = 13.
1 k(M m) and the frequency is v = = 2 2 Mm
k(M m) . Mm
Assume that a narrow tunnel is dug between two diametrically opposite points of the earth. Treat the earth as a solid sphere of uniform density. Show that if a particle is released in this tunnel, it will execute a simple harmonic motion. Calculate the time period of this motion.
Sol.
Consider the situation shown in figure. Suppose at an instant t the particle in the tunnel is at a distance x from the centre of the earth. Let us draw a sphere of radius x with its centre at the centre of the earth. Only the part of the earth within this sphere will exert a net attraction on the particle. Mass of this part is
4 3 x x3 3 M = M = 3 M. 4 R R 3 3 The force of attraction is, therefore, F=
G( x 3 / R 3 )Mm
The force constant is k =
T = 2
14.
m = 2 k
=
GMm
x. R3 x This force acts towards the centre of the earth. Thus, the resultant force on the particle is opposite to the displacement from the centre of the earth and is proportional to it. The particle, therefore, executes a simple harmonic motion in the tunnel with the centre of the earth as the mean position. 2
GMm R3
, so that the time period is
R3 GM
A simple pendulum of length 40 cm oscillates with an angular amplitude of 0.04 rad. Find (a) the time period, (b) the linear amplitude of the bob, (c) the speed of the bob when the string makes 0.02 rad with the vertical and (d) the angular acceleration when the bob is in momentary rest. Take g = 10 m/s2.
Sol.(a) The angular frequency is =
g/ =
10 m / s 2 = 5 s–1 0.4m
manishkumarphysics.in
Page # 15
Chapter # 12 the time period is
Simple Harmonic Motion
2 2 = = 1.26 s. 5 s 1
(b) (c)
Linear amplitude = 40 cm × 0.04 = 1.6 cm Angular speed at displacement 0.02 rad is = (5 s–1)
(d)
(0.04 )2 (0.02)2 rad = 0.17 rad/s.
where speed of the bob at this constant = (40 cm) × 0.175–1 = 6.8 cm/s. At momentary rest, the bob is in extreme position. Thus, the angular acceleration = (0.04 rad) (25 s–2) = 1 rad/s2.
15.
A simple pendulum having a bob of mass m undergoes small oscillations with amplitude 0. Find the tension in the string as a function of the angle made by the string with the vertical. When is this tension maximum, and when is it minimum ?
Sol.
Suppose the speed of the bob at angle is v. Using conservation of energy between the extreme position and the position with angle ,
1 mv2 = mg (cos – cos0). 2
.............(i)
As the bob moves in a circle path, the force towards the centre should be equal to mv2/. Thus, T = mg cos = mv2/. Using (i), T – mg cos = 2mg (cos – cos0) or, T = 3 mg cos – 2 mg cos0. Now cos is maximum at = 0 and decreases as || increases (for || < 90°). Thus, the tension is maximum when = 0 i.e., at the mean position and is minimum when = ± 0 i.e., at extreme positions. 16.
A simple pendulum is taken at a place where it separation from the earth’s surface is equal to the radius of the earth. Calculate the time period of small oscillations if the length of the string is 1.0m. Take g = 2 m/s2 at the surface of the earth.
Sol.
At a height R (radius of the earth) the acceleration due to gravity is GM
1 GM = g/4. (R R ) 4 R2 The time period of small oscillations of the simple pendulum is g=
2
=
T = 2 / g = 2
1.0 m 2 s 1 2 2 = 2 = 4s. m/s 4
17.
A simple pendulum is suspended from the ceiling of a car accelerating uniformly on a horizontal road. If the acceleration is a0 and the length of the pendulum is , find the time period of small oscillations about the mean position.
Sol.
We shall work in the car frame. As it is accelerated with respect to the road, we shall have to apply a psuedo force ma0 on the bob of mass m. manishkumarphysics.in Page # 16
Chapter # 12 Simple Harmonic Motion For mean position, the acceleration of the bob with respect to the car should be zero. If be the angle made by the string with the vertical, the tension, weight and the psuedo force will add to zero in this position.
Suppose, at some instant during oscillation, the string is further deflected by an angle so that displacement of the bob is x. Taking the components perpendicular to the string, component of T = 0, component of mg = mg sin ( + ) and component of ma0 = – ma0 cos ( + ). Thus, the resultant component F = m [g sin ( + ) –a0 cos ( + )] Expanding the sine and cosine and putting cos 1, sin = x/, we get
t h e
x F = m g sin a 0 cos ( g cos a 0 sin ) . ...........(i) At x = 0, the force F on the bob should be zero, as this is the mean position. Thus by (i), 0 = m[g sin – a0 cos ] ..........(ii)
giving
a0 tan = g
Thus,
sin =
a0
..........(iii)
a 02 g2
g cos =
...........(iv)
a 02 g2
Putting (ii), (iii) and (iv) in (i), F = m
g2 a 02
x
g2 a02 This is an equation of simple harmonic motion with time period or,
F = m 2 x, where 2 =
t=
2 = 2 2 (g a 02 )1/ 4
An easy working rule may be found out as follows. In the mean position, the tension, the weight and the psuedo force balance. From figure, the tension is T=
(ma 0 ) 2 (mg ) 2
T = a 02 g2 . m This plays the role of effective ‘g’. Thus the time period is or,
t = 2
= 2 2 . [g a 02 ]1/ 4 T /m
18.
A uniform meter stick is suspended through a small pin hole at the 10 cm mark. Find the time peiod of small oscillation about the point of suspension.
Sol.
Let the mass of the stick be m. The moment of inertia of the stick about the axis of rotation through the point of suspension is manishkumarphysics.in
Page # 17
Chapter # 12
Simple Harmonic Motion
m 2 + md2, 12 where = 1 m and d = 40 cm. =
The separation between the centre of mass of the stick and the point of suspension is d = 40 cm. The time period of this physical pendulum is
T = 2 mgd
= 2
1 = 2 12 0.16 / 4 s = 1.55 s.
m 2 md2 /(mgd) 12
19.
The moment of inertia of the disc used in a torsional pendulum about the suspension wire is 0.2 km-m2. If oscillates with a period of 2s. Another disc is placed over the first one and the time period of the system becomes 2.5 s. Find the moment of inertia of the second disc about the wire.
Sol.
Let the torsional constant of the wire be k. The moment of inertia of the first disc about the wire is 0.2 kg-m2. Hence the time period is
k
0.2 kg m 2 ........... (i) k When the second disc having moment of inertia 1 about the wire is added, the time period is 2s = 2
2.5 s = 2
= 2
0.2 kg m 2 1 k
..............(ii)
0.2 kg m 2 1 6.25 From (i) and (ii), = 4 0.2 kg m 2
This gives 1 0.11 kg m2. 20.
A uniform rod of mass m and length is suspended through a light wire of length and torsional constant k as shown in figure. Find the time period of the system makes (a) small oscillations in the vertical plane about the suspension point and (b) angular oscillations in the horizontal plane about the centre of the rod.
Sol.(a) The oscillations take place about horizontal line through the point of suspension and perpendicular to the plane of the figure. The moment of inertia of the rod about this line is
13 m 2 + m2 = m2. 12 12 The time period = 2 (b)
mg = 2
13 m 2 12 mg
= 2
13 12 g .
The angular oscillations take place about the suspension wire. The moment of inertia about this line is m2/ 12. The time period is
manishkumarphysics.in
Page # 18
Chapter # 12
Simple Harmonic Motion 2
21.
m 2 = 2 . 12 k k
A particle is subjected to two simple harmonic motions x1 = A1 sin t and x2 = A2 sin (t + /3). Find (a) the displacement at t = 0, (b) the maximum speed of the particle and (c) the maximum acceleration of the particle.
Sol.(a) At and
t = 0, x1 = A1 sin t = 0 x2 = A2 sin (t + /3)
A2 3 . 2 Thus, the resultant displacement at t = 0 is = A2 sin (/3) =
3 2 The resultant of the two motions is a simple harmonic motion of the same angular frequency . The amplitude of the resultant motion is x = x1 + x2 = A2
(b)
A= =
A 12 A 22 2A 1A 2 cos( / 3 ) A 12 A 22 A 1A 2 .
The maximum speed is umax = A = A 12 A 22 A 1A 2 (c)
The maximum acceleration is amax = A 2 = 2
22.
A12 A 22 A 1A 2 .
A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions, find the phase difference between the individual motions. HCV_Ch-12_WOE _22
,d d.k nks ljy vkorZ xfr;ksa ftuds vk;ke o vko`fÙk leku gS ds vUrxZr xfreku gSA ;fn ifj.kkeh vk;ke vyx&vyx xfr;ksa ds vk;ke ds cjkcj gS rks vyx&vyx xfr;ksa ds chp dykUrj Kkr djksA Sol.
Let the amplitudes of the individual motions be A each. The resultant amplitude is also A. If the phase difference between the two motions is , A= or, or, or,
=A
A 2 A 2 2A. A. cos 2(1 cos ) = 2A cos
2
1 = 2 2 = 2/3.
cos
EXERCISE 1.
A particle executes simple harmonic motion with an amplitude of 10 cm and time period 6 s. At t = 0 it is at position x = 5 cm going towards positive x-direction. Write the equation for the displacement x at time t. Find the magnitude of the acceleration of the particle at t = 4 s. ,d d.k 10 lseh vk;ke rFkk 6 lsd.M vkorZdky ds lkFk ljy vkorZ xfr dj jgk gSA /kukRed x fn'kk esa tkrs gq,] t = 0 ij bldh fLFkfr x = 5 lseh gSA le; t ij foLFkkiu x-ds fy;s lehdj.k fyf[k;sA t = 4 lsd.M ij
d.k ds Roj.k dk ifjeki Kkr dhft;sA
manishkumarphysics.in
Page # 19
Chapter # 12 1.
Simple Harmonic Motion 2 t , 11 cm/s2 x = (10 cm) sin 6 6s
2.
The position, velocity and acceleration of a particle executing simple hamonic motion are found to have magnitudes 2 cm, 1 m/s and 10 m/s2 at a certain instant Find the amplitude and the time period of the motion. ljy vkorZ xfr dj jgs d.k dh fdlh fof'k"V {k.k ij fLFkfr osx rFkk Roj.k ds ifjek.k Øe'k% 2 lseh, 1 eh- /ls- 10 eh-/ls-2 gSa xfr dk vk;ke rFkk vkorZdky Kkr dhft;sA 2. 4.9 cm, 0.28 s
3.
The particle executes simple harmonic motion with an amplitude of 10 cm. At what distance from the mean position are the kinetic and potential energies equal ? ,d d.k 10 lseh vk;ke ds lkFk ljy vkorZxfr dj jgk gSA ek/; fLFkfr ls fdl nwjh ij xfrt rFkk fLFkfrt ÅtkZ,¡
cjkcj gksxh A 3. 4.
5 2
The maximum speed and acceleration of a particle executing simple harmonic motion are 10 cm/s and 50 cm/s2. Find the position (s) of the particle when the speed is 8 cm/s. ljy vkorZxfr dj jgs d.k dh vf/kdre pky rFkk Roj.k 10 lseh/ls- ,oa 50 lseh/ls2.gSA tc d.k dh pky 8 lseh@
ls- gks rks d.k dh fLFkfr ¼ fLFkfr;k¡ ½ Kkr dhft;sA 4.
± 1.2 cm from the mean position
5.
A particle having mass 10 g oscillates according to the equation x = (2.0 cm) sin [(100 s–1) t+ /6]. Find (a) the amplitude, the time period and the spring constant (b) the position, the velocity and the acceleration at t = 0. 10 xzke nzO;eku dk d.k lehdj.k x = (2.0 cm) sin [(100 s–1) t+ /6] ds vuqlkj nksy u dj jgk gSA Kkr dhft;s% (a) vk;ke] vkorZdky rFkk fLizax fu;rkad (b) t = 0 ij fLFkfr] osx rFkk Roj.k 5. (a) 2.0 cm, 0.063 s, 100 N/m (b) 1.0 cm, 1.73 m/s, 100 m/s2
6.
The equation of motion of a particle started at t = 0 is given by x = 5 sin (20 t + /3) where x is in centimetre and t in second. When does the particle (a) first come to rest (b) first have zero acceleration (c) first have maximum speed ? t = 0 ij xfr izkjEHk djus okys ,d d.k dh xfr dh lehdj.k x = 5 sin (20 t + /3) }kjk O;Dr dh tkrh gS] tgk¡ x lseh esa rFkk t lsd.M esa gSA d.k (a) igyh ckj dc fojkekoLFkk esa vk;sxk (b) dk Roj.k igyh ckj dc 'kwU; gksxk (c) dh pky igyh ckj dc vf/kdre gksxh \ 6.
7.
(a)
s 120
(b)
s 30
(c)
s 30
Consider a particle moving in simple harmonic motion according to the equation x = 2.0 cos (50 t + tan–1 0.75) where x is in centimetre and t in second. The motion is started at t = 0. (a) When does the particle come to rest for the first time ? (b) When does the acceleration have its maximum magnitude for the first time ? (c) When does the particle come to rest for the second time ?
,d d.k fuEufyf[kr lehdj.k ds vuqlkj ljy vkorZ xfr dj jgk gSA x = 2.0 cos (50 t + tan–1 0.75) ;gk¡ x lseh esa rFkk t lsd.M esa gSA xfr t = 0 ij
izkjEHk gksrh gSA (a) d.k igyh ckj fojkekoLFkk esa dc vk;sxk\ (b) igyh ckj Roj.k dk ifjek.k vf/kdre dc gksxk\ (c) d.k nwljh ckj fojkekoLFkk esa dc vk;sxk\ 7. 8.
(a) 1.6 × 10–2 s (b) 1.6 × 10–2 s (c) 3.6 × 10–2 s
Consider a simple harmonic motion of time period T. Calculate the time taken for the displacement to change value from half the amplitude to the amplitude. T vkorZdky ls gks jgh ljy vkorZ xfr ij fopkj dhft;sA foLFkkiu dk eku vk;ke ds vk/ks ls vk;ke rd ifjofrZr manishkumarphysics.in
Page # 20
Chapter # 12
Simple Harmonic Motion
gksus esa yxs le; dh x.kuk dhft;sA 8. 9.
T/6
The pendulum of a clock is replaced by a spring - mass system with the spring having spring constant 0.1 N/ m. What mass should be attached to the spring ? ,d ?kM+h dk yksyd] fLizax&nzO;eku fudk; ls izfrLFkkfir dj fn;k x;k gS] fLizax dk fu;rkad 0.1 U;wVu@eh- gSA fLizax
ls fdruk nzO;eku tksM+uk pkfg;s\ 9. 10.
10 g
A block suspended from a vertical spring is in equilibrium. Show that the extension of the spring equals the length of an equivalent simple pendulum i.e., a pendulum having frequency same as that of the block.
,d m/okZ/kj fLizax ls yVdk;k x;k xqVdk lkE;koLFkk esa gSA O;Dr dhft;s fd fLizax dk izlj.k] ,d rqY; ljy yksyd dh yEckbZ ds cjkcj gS vFkkZr~ CykWd dh vko`fÙk ds cjkcj vko`fÙk okys yksyd dh yEckbZA 11.
A block of mass 0.5 kg hanging from a vertical spring executes simple harmonic motion of amplitude 0.1 m and time period 0.314 s. Find the maximum force exerted by the spring on the block. 11. 25 N ,d m/okZ/kj fLizax ls yVdk;k x;k 0.5 fdxzk nzO;eku dk xqVdk ljy vkorZ xfr dj jgk gSA ftldk vk;ke 0.1 eh- rFkk vkorZdky 0.314 lsd.M gSA fLizax }kjk xqVds ij yxk;k x;k vf/kdre cy Kkr dhft;sA
12.
A body of mass 2 kg suspended through a vertical spring executes simple harmonic motion of period 4s. If the oscillations are stopped and the body hangs in equilibrium, find the potential energy stored in the spring. ,d m/okZ/kj fLizax ls yVdk;h x;h 2 fdxzk nzO;eku okyh oLrq] ljy vkorZ xfr dj jgh gS] bldk vkorZdky 4 lsd.M
gSA ;fn nksyu jksd fn;s tk;s rFkk oLrq lkE;koLFkk esa yVdh jgs] fLizax esa lafpr fLFkfrt ÅtkZ Kkr dhft;sA 12. 13.
40 J
A spring stores 5 J of energy when stretched by 25 cm. It is kept vertical with the lower end fixed. A block fastened to its other end is made to undergo small oscillations. If the block makes 5 oscillations each second, what is the mass of the block ? ,d fLizax dks 25 lseh [khapus ij blesa 5 twy ÅtkZ lafpr gksrh gSA bldks m/okZ/kj j[kk x;k gS rFkk bldk fupyk
fljk fLFkj gSA nwljs fljs ij ,d xqVdk fpidk;k x;k gS tks vYi vk;ke okys nksyu dj jgk gSA ;fn xqVdk izR;sd lsd.M esa 5 nksyu dj jgk gS] xqVds dk nzO;eku fdruk gS\ 13. 14.
0.16 kg
A small block of mass m is kept on a bigger block of mass M which is attached to a vertical spring of spring constant k as shown in the figure. The system oscillates vertically. (a) Find the resultant force on the smaller block when it is displaced through a distance x above its equilibrium position. (b) Find the normal force on the smaller block at this position. When is this force smallest in magnitude? (c) What can be the maximum amplitude with which the two blocks may oscillate together? m nzO;eku okyk ,d NksVk xqVdk] M nzO;eku okys cM+s xqVds ij j[kk gqvk gS] tks fp=kkuqlkj ,d k cy fu;rkad okyh m/okZ/kj fLizax ls tqM+k gqvk gSA fudk; m/okZ/kj nksyu dj jgk gSA (a) tc bldks lkE;koLFkk ls Åij dh vksj x nwjh rd foLFkkfir fd;k tkrk gS] NksVs xqVds ij ifj.kkeh cy Kkr dhft;sA (b) bl fLFkfr esa NksVs xqVds ij vfHkyEcor~ cy Kkr dhft;sA ;g cy U;wure ifjek.k dk dc gksxk\ (c) vk;ke dk vf/kdre eku fdruk gks ldrk gS fd nksuksa
xqVds ,d lkFk nksyu djrs jgsA
14. 15.
(a)
mkx mkx (M m) (b) mg – , at the highest point (c) g Mm Mm k
The block of mass m1 shown in figure is fastened to the spring and the block of mass m2 is placed against it. (a) Find the compression of the spring in the equilibrium position. (b) The blocks are pushed a further distance (2/k) (m1 + m2) g sin against the spring and released. Find the position where the two blocks separate. (c) What is the common speed of blocks at the time of separation? fp=k esa iznf'kZr fd;k x;k m1 nzO;eku dk CykWd fLizax ls fpidk;k x;k gS rFkk m2 nzO;eku okyk CykWd blls fVdk dj j[kk x;k gSA (a) lkE;koLFkk esa fLizax dk lEihM+u Kkr dhft;sA (b) CykWdksa dks (2/k) (m1 + m2) g sin nwjh rd vksj foLFkkfir djds NksM+ fn;k tkrk gSA og fLFkfr Kkr dhft;s] tgk¡ nksuksa CykWd vyx gks tk;saxsA (c) vyx gksrs
le; nksuksa CykWdksa dh mHk;fu"B pky fdruh gksxh\
manishkumarphysics.in
Page # 21
Chapter # 12
15. 16.
Simple Harmonic Motion
(a)
(m1 m 2 ) g sin (b) when the spring acquires its natural length (c) k
In figure k = 100 N/m, M = 1 kg and F = 10 N. (a) Find the compression of the spring in the equilibrium position. (b) A sharp blow by some external agent imparts a speed of 2 m/s to the block towards left. Find the sum of the potential energy of the spring and the kinetic energy of the block at this instant. (c) Find the time period of the resulting simple harmonic motion. (d) Find the amplitude. (e) Write the potential energy of the spring when the block is at the left extreme. (f) Write the potential energy of the spring when the block is at the right extreme. The answers of (b), (e) and (f) are different. Explain why this does not violate the principle of conservation of energy. fp=k esa k = 100 U;wVu/eh-, M = 1 fdxzk rFkk F = 10 U;wVu gSA (a) lkE;koLFkk esa fLizax dk lEihM+u Kkr dhft;sA (b) fdlh cká ;qfDr }kjk ,dne yxk;s x;s /kDds ds dkj.k CykWd cka;h vksj 2 eh-/ls- dh pky izkIr dj ysrk gSA bl {k.k ij fLizax dh fLFkfrt ÅtkZ rFkk CykWd dh xfrt ÅtkZ dk ;ksx dhft;sA (c) ifj.kkeh ljy vkorhZ xfr dk vkorZdky Kkr dhft;sA (d) vk;ke Kkr dhft;s (e) tc CykWd cka;h vksj vafre fLFkfr esa gks rks fLizxa dh fLFkfrt ÅtkZ fyf[k;sA (f) CykWd tc nk;ha vksj vafre fLFkfr esa gks fLizax dh fLFkfrt ÅtkZ fyf[k;sA The answers of (b), (e) and (f) are different. Explain why this does not violate the principle of conservation of energy.
16. (a) 10 cm 17.
3 (m1 m 2 )g sin k
(b) 2.5 J
(c) /5 s
(b) 20 cm
(e) 4.5 J
(f) 0.5 J
Find the time period of the oscillation of mass m in figures a, b, c. What is the equivalent spring constant of the pair of springs in each case ? fp=k (a, b, c) esa iznf'kZr m nzO;eku ds nksyu dk vkorZdky Kkr dhft;sA izR;sd fLFkfr esa fLizaxksa ds tksM+s dk rqY;
fLizax fu;rkad fdruk gksxk\
(a) 17. 18.
(b)
(c)
m (k1 k 2 ) m m (a) 2 k k (b) 2 k k (c) 2 k1k 2 1 2 1 2
The spring shown in figure is unstretched when a man starts pulling on the cord. The mass of the block is M. If the man exerts a constant force F, find (a) the amplitude and the time period of the motion of the block (b) the energy stored in the spring when the block passes through the equilibrium position and (c) the kinetic energy of the block at this position.
tc ,d O;fDr Mksjh [khapuk izkjEHk djrk gS rks fp=k esa iznf'kZr fLizxa fcuk [khaph gqbZ fLFkfr esa gSA CykWd dk nzO;eku M gSA ;fn O;fr }kjk yxk;k x;k fu;r cy F gS] Kkr dhft;s A (a) CykWd dh xfr dk vk;ke rFkk vkorZdky (b) tc CykWd lkE;koLFkk ls xqtjrk gS rks fLizax esa lafpr ÅtkZ rFkk (c) bl fLFkfr esa CykWd dh xfrt ÅtkZ
18. 19.
(a)
F M F2 , 2 , (b) k k 2k
(c)
F2 2k
A particle of mass m is attatched to three springs A, B and C of equal force constant k as shown in figure. If the particle is pushed slightly against the spring C and released, find the time period of oscillations. tSlkfd fp=k esa iznf'kZr fd;k x;k gSA leku cy fu;rkad k okyh rhu fLizaxksa A, B rFkk C ls m nzO;eku dk ,d d.k manishkumarphysics.in Page # 22
Chapter # 12
Simple Harmonic Motion
tqM+k gqvk gSaA ;fn d.k dks fLizax C ds fo:) FkksMk lk foLFkkfir djds NksM+ fn;k tkrk gS] nksyu dk vkorZdky Kkr dhft;sA
19. 20.
m 2k
Repeat the previous exercise if the angle between each pair of springs is 120° initially. ;fn izR;sd fLizax ;qXe ds chp izkjfEHkd dks.k 120° gks rks fiNys iz'u% dks nksgjkb;sA 20.
21.
2
2
2m 3k
The springs shown in the figure are all unstretched in the beginning when a man starts pulling the block. The man exerts a constant force F on the block. Find the amplitude and the frequency of the motion of the block.
tc dksbZ O;fr CykWd dks [khpuk 'kq: djrk gS rks fp=k esa iznf'kZr leLr fLizaxsa] izkjEHk esa fcuk [khph gbZ fLFkfr esa gSA O;fDr CykWd ij ,d fu;r cy F yxkrk gSA CykWd dh xfr dk vk;ke rFkk vko`fr Kkr dhft;sA
21.
22.
F(k 2 k 3 ) 1 k 1 k 2 k 2 k 3 k 3 k 1 , 2
k1 k 2 k 2 k 3 k 3 k 1 M(k 2 k 3 )
Find the elastic potential energy stored in each spring shown in figure, when the block is in equilibrium. Also find the time period of vertical oscillation of the block.
fp=k esa iznf'kZr CykWd tc lkE;koLFkk esa gS] izR;sd fLizax esa lafpr izR;kLFk fLFkfrt ÅtkZ Kkr dhft;sA
22. 23.
M2 g2 M2 g2 M2 g2 , and from above, time period = 2 2k1 2 k2 2 k3
1 1 1 M k k k 2 3 1
The string, the spring and the pulley shown in figure are light. Find the time period of the mass m. fp=k esa Mksjh] fLizax rFkk f/kjuh ux.; nzO;eku dh gSA m nzO;eku dk vkorZdky Kkr dhft;sA
23.
2
m k manishkumarphysics.in
Page # 23
Chapter # 12 Simple Harmonic Motion 24. Solve the previous problem if the pulley has a moment of inertia about its axis and the string does not slip over it. fiNys iz'u dks gy dhft;s] ;fn f?kjuh dk bldh v{k ds ifjr% tMRo vk?kw.kZ gS rFkk Mksjh bl ij fQlyrh ugh
gSA
24.
25.
2
(m / r 2 ) k
Consider the situation shown in figure. Show that if the blocks are displaced slightly in opposite directions and released, they will execute simple harmonic motion. Calculate the time period.
fp=k esa iznf'kZr O;oLFkk ij fopkj dhft;sA ;fn CykWdksa dks foijhr fn'kkvksa esa FkksM+k lk foLFkkfir djds NksM+ fn;k rks O;Dr dhft;s fd os ljy vkorZ xfr djsxasA vkorZdky dh x.kuk dhft;s A
25.
26.
m 2 2 k
A rectangular plate of sides a and b is suspended from a ceiling by two parallel strings of length L each (figure). The separation between the strings is d. The plate is displaced slightly in its plane keeping the strings tight. Show that it will excute simple harmonic motion. Find the time period. Nr ls yVrh gqbZ nks lekukarkj Mksfj;ksa ls ,d vk;rkdkj IysV yVdkbZ x;h gS] bldh Hkqtk,¡ a rFkk b gSA izR;sd Mksjh dh yEckbZ L gSA Mksfj;ksa ds chp dh nqjh d gSA Mksfj;kas dks ruk gqvk j[krs gq, IysV dks blds ry esa FkksM+k lk foLFkkfir
fd;k tkrk gSA O;Dr dhft;s fd ;g ljy vkorZ xfr djsxhA vkorZdky Kkr dhft;sA
26.
27.
2p
L g
A 1 kg block is executing simple harmonic motion of amplitude 0.1 m on a smooth horizontal surface under the restoring force of a spring of spring constant 100 N/m. A block of mass 3 kg is gently placed on it at the instant it passes through the mean position. Assuming that the two blocks move together, find the frequency and the amplitude of the motion. 100 U;wVu@ eh- cy fu;rkad okyh fLiazx ds izR;ku;u cy ds izHkko esa ,d fpduh {ksfrt lrg ij j[kk gqvk 1 fdxzk nzO;eku dk CykWd ljy vkorZ xfr dj jgk gSA tc ;g ek/; fLFkfr ls xqtjrk gS] blds Åij ,d 3 fdxzk nzO;eku
dk CykWd /khjs ls j[k fn;k tkrk gSA ;g ekurs gq, fd nksuks CykWd ,d lkFk xfr djrs gS] xfr dh vko`fr ,oa vk;ke Kkr dhft;s A
27. 28.
5 Hz, 5 cm 2
The left block in figure moves at a speed v towards the right block placed in equilibrium. All collisions to take place are elastic and the surface are frictionless. Show that the motions of the two blocks are periodic. Find the time period of these periodic motions. Neglect the widths of the blocks. fp=k esa iznf'kZr ck;ka CykWd] lkE;koLFkk esa fLFkr nk;sa CykWd dh vksj v pky ls xfr'khy gSA gksus okyh leLr VDdjs
izR;kLFk gS rFkk lrgsa ?k"kZ.k jfgr gSA O;Dr dhft;s fd nksuksa Cykdks dh xfr vkorhZ gksxhA budh vkorhZ xfr;ksa dk vkorZdky Kkr dhft;s] CykWdks dh pkSMkbZ;k¡ ux.; eku yhft;s A manishkumarphysics.in
Page # 24
Chapter # 12
Simple Harmonic Motion
m 2L k v Find the time period of the motion of the particle shown in figure. Neglect the small effect of the bend near the bottom. 28.
29.
fp=k esa iznf'kZr d.k dh xfr dk vkorZdky Kkr dhft;sA iSnsa ds fudV eksM+ dk izHkko ux.; eku yhft;sA
29. 30.
All the surfaces shown in figure are frictionless. The mass of the car is M, that of the block is m and the spring has spring constant k. Initially, the car and the block are at rest and the spring is stretched through a length x0 when the system is released. (a) Find the amplitudes of the simple harmonic motion of the block and of the car as seen from the road. (b) Find the time period (s) of the two simple harmonic motions. fp=k esa iznf'kZr leLr lrgs ?k"kZ.k jfgr gSA dkj dk nzO;eku M, CykWd dk nzO;eku m rFkk fLizax dk fLizax fu;rkad k gS A izkjEHk esa tc fudk; dks eqDr fd;k x;k dkj rFkk CykWd fojkekoLFkk esa gS ,oa fLizax x0 yEckbZ ls [khph gqbZ gSA (a)lM+d ls ns[kus ij CykWd rFkk dkj ds vk;keksa dk eku Kkr dhft;s (b) nksuksa ljy vkorZ xfr;ksa dk vkorZdky (ds vkorZdky ) Kkr dhft;sA
30.
31.
0.73 s
(a)
mM Mx 0 mx 0 , (b) 2 k (M m) Mm Mm
A uniform plate of mass M stays horizontally and symmetricaly on two wheels rotating in opposite directions (figure). The separation between the wheels is L. The friction coefficient between each wheel and the plate is µ. Find the time period of oscillation of the plate if it is slightly displaced along its length and released. foifjr fn'kkvksa esa ?kwf.kZr nks ifg;ksa ij M nzO;eku dh ,d le:i IysV {ksfrt :i ls j[kh gqbZ gSA ifg;s ds e/; varjky LgSA izR;sd ifg;s rFkk IysV ds e/; ?k"kZ.k xq.kkad µ gS A ;fn IysV dks bldh yEckbZ ds vuqfn'k FkksM+k lk foLFkkfir
djds NksM+ fn;k tk;s rks blds nksyu dk vkorZdky Kkr dhft;sA
31.
32.
2
2µg
A pendulum having time period equal to two seconds is called a seconds pendulum. Those used in pendulum clocks are of this type. Find the length of a seconds pendulum at a place where g = 2 m/s2.
nks lsd.M vkorZdky okyk yksykd] ^^ lsd.M+ yksyd ^^ dgykrk gSA ?kfM+;ksa esa bl izdkj ds yksyd iz;qDr fd;s tkrs gSA ;fn fdlh LFkku ij g = 2 m/s2 gS] ogkW ij lsd.M yksyd dh yEckbZ Kkr dhft;s A 32. 33.
1m
The angle made by the string of a simple pendulum with the vertical depends on time as =
sin [ s–1) 90
t]. Find the length of the pendulum if g = 2 m/s2.
ljy yksyd dh Mksjh dk m/okZ/kj ls dks.k le; ij =
sin [ s–1) t] ds 90
manishkumarphysics.in
vuqlkj fuHkZj djrk gSA ;fn g Page # 25
Chapter # 12
Simple Harmonic Motion
= m/s gS rks 33. 1m 2
34.
2
yksyd dh yEckbZ Kkr dhft;s\
The pendulum of a certain clock has time period 2.04 s. How fast or slow does the clock run during 24 hours ? fdlh fof'k"V ?kM+h ds yksyd dk vkorZdky 2.04 ls- gSA 24 ?kaVs i'pkr~ ;g fdlh rst vFkok /kheh pysxh ? 34.
28.8 minutes slow
35.
A pendulum clock giving correct time at a place where g = 9.8 m/s2 is taken to another place where it loses 24 seconds during 24 hours. Find the value of g at this new place. fdlh LFkku ij tgk¡ g = 9.8 m/s2 gSA yksyd ?kMZ lgh le; n'kkZrh gSA bldks fdlh vU; LFkku ij ys tkus ij ;g 24 ?kaVks esa 24 lsd.M ihNs jg tkrh gSA bl u;s LFkku ij g dk eku Kkr dhft;sA 35. 9.795 m/s2
36.
A simple pendulum is constructed by hanging a heavy ball by a 5.0 m long string. It undergoes small oscillations. (a) How many oscillations does it make per second ? (b) What will be the frequency if the system is taken on the moon where acceleration due to gravitation of the moon is 1.67 m/s2. ,d 5.0 m yEch Mksjh ls ,d Hkkjh xasn yVdkdj ljy yksyd cuk;k x;k gSA ;g vYi vk;ke ds nksyu dj jgk gSA (a) ;g izfr lsd.M fdrus nksy u djsxk ? (b) ;fn bl fudk; dks pUnzek ij ys tk;k tk;s tgk¡ xq:Roh; Roj.k dk eku 1.67 m/s2 gS] bldh vko`fr fdruh gksxh \ 36.
37.
(a) 0.70/ (b) 1/(2/ 3 ) Hz
The maximum tension in the string of an oscillating pendulum is double of the minimum tension. Find the angular amplitude.
nksyu dj jgs ,d yksyd dh Mksjh esa vf/kdre ruko] blds U;wure ruko dk nqxuk gSA dks.kh; vk;ke Kkr dhft;sA 37. 38.
cos–1 (3/4)
A small block oscillates back and forth on a smooth concave surface of radius R (figure). Find the time period of small oscillation. R f=kT;k dh ,d fpduh vory lrg ij ,d NksVk CykWd vkxs&ihNs nksyu dj jgk gSA (fp=k) vYi vk;ke ds nksyuksa
dk vkorZdky Kkr fdft;sA
38. 39.
2 R / g
A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period. cgqr vf/kd f=kT;k R dh ,d [kqjnjh vory lrg ij m nzO;eku dh xksykdkj xasn fcuk fQlys yq<+d jgh gSA ;g
fuEure fcUnq ds vklikl vYi vk;ke ds nksyu dj jgh gSA vkorZdky Kkr dhft;sA 39.
2
7(R r ) 5g
40.
A simple pendulum of length 40 cm is taken inside a deep mine. Assume for the time being that the mine is 1600 km deep. Calculate the time period of the pendulum there. Radius of the earth = 6400 km. 40 lseh yEck ,d ljy yksyd] xgjh [kku ds vUnj ys tk;k tkrk gSA dqN le; ds fy;s eku yhft;s fd [kku dh xgjkbZ 1600 fdeh gSA ogk¡ ij yksyd ds vkorZdky dh x.kuk dhft;saA i`Foh dh f=kT;k = 6400 fdeh A 40. 1.47 s
41.
Assume that a tunnel is dug across the earth (radius = R) passing through its centre. Find the time a particle takes to cover the length of the tunnel if (a) it is projected into the tunnel with a speed of
gR (b) it is released
from a height R above the tunnel (c) it is thrown vertically upward along the length of tunnel with a speed of gR .
eku yhft;s fd i`Foh (f=kT;k = R) ds vkjikj ,d lqjax [kksnh xbZ gS tks dsUnz ls gksdj xqtjrh gSA lqjax dh lEiw.kZ yEckbZ r; djus esa fdlh d.k dks yxus okyk le; Kkr dhft;s] ;fn (a) bldks lqjax es gR pky ls iz{ksfir fd;k tk;sA (b) bldks lqjax ds Åij R Å¡pkbZ ls NksM+k tk;sA (c) bldks lqjax dh yEckbZ ds vuqfn'k Åij dh vksj gR pky manishkumarphysics.in
Page # 26
Chapter # 12
Simple Harmonic Motion
ls iz{ksfir fd;k tk;s A 41. 42.
2
R g in each case
Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance R/2 from the earth’s centre where R is the radius of the earth. The wall of the tunnel is frictionless. (a) Find the gravitational force exerted by the earth on a particle of mass m placed in the tunnel at a distance x from the centre of the tunnel. (b) Find the component of this force along the tunnel and perpendicular to the tunnel. (c) Find the normal force exerted by the wall on the particle. (d) find the resultant force on the particle. (e) Show that the motion of the particle in the tunnel is simple harmonic and find the time period. eku yhft;s fd i`Foh ds dsUnz ls R/2 yEcor~ nwjh ij fLFkr] thok ds vuqfn'k ,d lqjax [kksnh tkrh gS] ;gk¡ R i`Foh dh f=kT;k gSA (a) lqjax ds dsUnz ls x nwjh ij fLFkr m nzO;eku ds d.k ij i`Foh ds dkj.k yxus okys xq:Roh; cy dk eku Kkr dhft;sA (b) lqjax ds vuqfn'k ,oa lqjax ds yEcor~ bl cy ds ?kVd Kkr dhft;sA (c) nhokj }kjk d.k ij yxk;k x;k vfHkyEcor~ cy Kkr fdft;sA (d)nhokj ij ifj.kkeh cy Kkr dhft;sA (e) O;Dr dhft;s fd lqjax esa
d.k dh xfr ljy vkorhZ gS] bldka vkorZdky Hkh Kkr dhft;sA 42. 43.
(a)
GMm R
3
x 2 R 4 / 4 (b)
GMm R
3
x,
GMm 2R
2
(c)
GMm 2R
2
, (d)
GMm R3
x (e) 2 R 3 /(GM)
A simple pendulum of length is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator (a) is going up with an acceleration a0 (b) is going down with an acceleration a0 and (c) is moving with a uniform velocity. ,d fy¶V dh Nr ls yEckbZ dk ljy yksyd yVdk;k x;k gSA vYi vk;ke ds nksyuksa dk vkorZdky Kkr dhft;s] ;fn fy¶V (a) Åij dh vksj a0 Roj.k ls xfr'khy gSaA (b) uhps dh vksj a0 Roj.k ls xfr'khy gS] rFkk (c) ,d leku osx
ls xfr'khy gSA 43. 44.
(a) 2
g a 0 (b) 2
(c) 2 g a0 g
A simple pendulum of length 1 feet suspended from the ceiling of an elevator takes /3 seconds to complete one oscillation. Find the acceleration of the elevator. ,d fy¶V dh Nr ls yVdk;k x;k 1 QqV yEckbZ dk ljy yksyd /3 lsd.M esa ,d nksyu iwjk dj ysrk gSA fy¶V
dk Roj.k Kkr dhft;sA 44. 45.
4 feet/s2 upwards
A simple pendulum fixed in a car has a time period of 4 seconds when the car is moving uniformly on a horizontal road. When the accelerator is pressed, the time period changes to 3.99 seconds. Making an approximate analysis, find the acceleration of the car. ,d dkj dh Nr ls yVdk;s x;s ljy yksyd dk vkorZdky 4 lsd.M gS] tcfd dkj {ksfrt lMd ij ,d leku :i ls xfr'khy gSA tc ,DlhysVj nck;k tkrk gSA bldk vkorZdky ifjofrZr gksdj 3.99 lsd.M gks tkrk gSA vuqekfur
fo'ys"k.k djds] dkj dk Roj.k Kkr dhft;s A 45. 46.
g/10
A simple pendulum of length is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation. r f=kT;k dh ,d {ksfrt o`Ùkkdkj lM+d ij ,d dk v pky ls xfr'khy gS] bldh Nr ls yEckbZ dk ,d ljy yksyd yVdk;k x;k gSA (a) tc ;g dkj ds lkis{k fLFkj gS] Mksjh esa ruko Kkr dhft;sA (b) vYi vk;ke okys nksyuksa dk
vkorZdky Kkr dhft;s A
1/ 2
47.
2 v 46. (a) ma (b) 2 / a where a = g 2 r The ear-ring of a lady shown in figure. has a 3 cm long light suspension wire. (a) Find the time period of small oscillations if the lady is standing on the ground. (b) The lady now sits in a merry-go-round moving at 4 m/s in a circle of radius 2 m. Find the time period of small oscillations of the ear-ring. fp=k esa iznf'kZr L=kh ds dku dk >qeds esa 3 lseh yEck ,d gYdk rkj gSA (a) ;fn L=kh tehu ij [kM+h gqbZ gS rks vYi vk;ke ds nksyuksa dk vkorZdky Kkr fdft;s A (b) ;fn ;g L=kh tc ^^ esjh&xks&jkm.M^^ esa cSB tkrh gS tks 4 eh-@
ls pky ls py jgk gSA >qeds ds vYi vk;ke ds nksyuksa dk vkorZdky Kkr dhft;sA manishkumarphysics.in
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Chapter # 12
47. 48.
(a) 0.34s (b) .30 s
Find the time period of small oscillations of the following systems. (a) A metre stick suspended through the 20 cm mark. (b) A ring of mass m and radius r suspended through a point on its perphery. (c) A uniform square plate of edge a suspended through a corner. (d) A uniform disc of mass m and radius r suspended through a point r/2 away from the centre. fuEu fudk;ksa ds fy;s vYi vk;ke ds nksyuksa dk vkorZdky Kkr dhft;s % (a)ehVj iSekuk ftls 20 lseh fpUg ij yVdk;k x;k gSA (b) r f=kT;k o m nzO;eku okyh oy;] ftls bldh ifjf/k ij fLFkr fcUnq ls yVdk;k x;k gSA (c) a Hkqtk okyh le:i oxkZdkj ifVV~dk ftls ,d dksus ls yVdk;k x;k gSA (d) m nzO;eku rFkk r f=kT;kokyh ,d pdrh ftls dsUnz ls r/2 nwjh ij fLFkr fcUnq ls yVdk;k x;k gSA 48.
49.
Simple Harmonic Motion
(a) 1.51 s (b) 2
2r g (c) 2
8a (d) 2 3g
3r 2g
A uniform rod of length is suspended by an end and is made to undergo small oscillations. Find the length of the simple pendulum having the time period equal to that of the rod. yEckbZ dh ,d le:i NM+ dks ,d fljs ls yVdk dj vYi vk;ke ds nksyu djok;s tkrs gSA ml ljy yksyd dh
yEckbZ Kkr dhft;sA ftldk vkorZdky NM+ ds vkorZdky ds rqY; gSA 49. 50.
2/3
A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period ? r f=kT;k dh ,d le:i pdrh esa ,d fNnz djds] blls pdrh dks yVdk;k tkuk gSA vYi vk;ke ds nksyuksa ds fy;s
pdrh dk U;wure lEHko vkorZdky Kkr dhft;sA U;wure vkorZdky ds fy;s fNnz dh dsUnz ls nqjh fdruh gksxh\ 50.
51.
2
r 2 , r/ 2 g
A hollow sphere of radius 2 cm is attached to an 18 cm long thread to make a pendulum. Find the time period of oscillation of this pendulum. How does it differ from the time period calculated using the formula for a simple pendulum ? yksyd cukus ds fy;s 2 lseh f=kT;k dk ,d [kks[kyk xksyk] 18 lseh yEcs /kkxs ls tksMk x;k gSA bl yksyd dk vkorZdky
Kkr dhft;sA ;g ljy yksyd dk vkorZdky ds lw=k dh lgk;rk ls dh xbZ x.kuk ls fdl izdkj fuEu gksxk \ 51.
0.89 s, it is about 0.3% larger than the calculated value
52.
A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude 2° and time period 2s. Find (a) the radius of the circular wire, (b) the speed of the particle farthest away from the point of suspnsion as it goes through its mean position, (c) the acceleration of this particle as it goes through its mean position and (d) the acceleration of this particle when it is at an exetreme position. Take g = 2 m/s2. nhokj ij yxh gqbZ dhy ls yVdk;k x;k ,d cUn o`Ùkkdkj rkj 2° vk;ke rFkk 2 ls.vkorZdky ds nksyu dj jgk gSA Kkr dhft;s &(a) o`Ùkkdkj rkj dh f=kT;k (b) tc ;g ek/; fLFkfr ls xqtjrk gS] rks fuyEcu fcUnq ls lokZf/kd nwjh okys d.k dh pky (c) bl d.k dk Roj.k tc ;g ek/; fLFkfr ls xqtjrk gS rFkk (d) tc ;g pje fLFkfr esa gksrk gS rks bl d.k dk Roj.kA g = 2 m/s2 eku yhft;sA 52. (a) 50 cm (b) 11 cm/s (c) 1.2 cm/s2 towards the point of suspension (d) 34 cm/s2 towards the mean position
53.
a uniform disc of mass m and radius r is suspended through a wire attached to its centre. If the time period of the torsional oscillations be T, what is the torsional constant of the wire. m nzO;eku o r f=kT;k okyh ,d le:i pdrh] blds dsUnz ls tqM+s gq, rkj ls yVdk;h x;h gS A ;fn bl ejksM+h nksyd dk vkorZdky T gS] rks rkj dk ,aBu fdruk gS\ manishkumarphysics.in Page # 28
Chapter # 12
53. 54.
Simple Harmonic Motion
22mr 2 T2
Two small balls, each of mass m are connected by a light rigid rod of length L. The system is suspended from its centre by a thin wire of torsional constant k. The rod is rotated about the wire through an angle 0 and released. Find the tension in the rod as the system passes through the mean position. nks NksVh xSans] izR;sd dk nzO;eku m gS] L yEckbZ dh ,d gYdh ,oa n`< NM+ ds fljs tqMs gq, gSA bl fudk; dks dsUnz ls] k ,aBu okys rkj ls yVdk;k x;k gSA NM+ dks rkj ds ikfjr% 0 dks.k ls ?kqekdj NksM+ fn;k tkrk gSA tc fudk;
ek/; fLFkfr ls xqtjrk gS] NM+ esa ruko Kkr dhft;sA
54.
55.
k 2 04 2 m 2 g2 L
1/ 2
A particle is subjected to two simple harmonic motions of same time period in the same direction. The amplitude of the first motion is 3.0 cm and that of the second is 4.0 cm. Find the resultant amplitude if the phase difference between the motions is (a) 0°, (b) 60°, (c) 90°.
,d d.k ij ,d dh fn'kk esa leku vkorZdky okyh nks ljy vkorZ xfr;k¡ vkjksfir dh tkrh gSA izFke xfr dk vkorZdky 3.0 lseh rFkk nqljh dk 4.0 lseh gSA ifj.kkeh vk;ke Kkr dhft;s] ;fn xfr;ksa ds e/; dykUrj gS % (a) 0°, (b) 60°, (c) 90°. 55. (a) 7.0 cm (b) 6.1 cm (c) 5.0 cm 56.
Three simple harmonic motions of equal amplitudes A and equal time periods in the same direction combine. The phase of the second motion is 60° ahead of the first and the phase of the third motion is 60° ahead of the second. Find the amplitude of the resultant motion. ,d gh fn'kk esa leku vk;ke A rFkk leku vkorZdky okyh rhu vkorZ xfr;k¡ la;ksftr dh tkrh gSA nwljh xfr izFke dh rqyuk esa dyk esa 60° dks.k ls vkxs gS rFkk rhljh xfr] nwljh dh rqyuk esa dyk esa 60° dks.k ls vkxs gSA ifj.kkeh
xfr dk vk;ke Kkr dhft;s 56. 57.
2A
A particle is subjected to two simple harmonic motions given by x1 = 2.0 sin (100 t) and x2 = 2.0 sin (120 t + /3) where x is in centimeter and t in second. Find the displacement of the particle at (a) t = 0.0125, (b) t = 0.025.
,d ij vkjksfir nks ljy vkorZ xfr;k¡ fuEukuqlkj xfr dk vk;ke Kkr dhft;s x1 = 2.0 sin (100 t) rFkk x2 = 2.0 sin (120 t + /3) tgk¡ x lseh esa rFkk t lsd.M esa gSA d.k dk foLFkkiu Kkr dhft;s (a) t = 0.0125, (b) t = 0.025. 57. 58.
(a) – 2.41 cm
(b) 0.27 cm
A particle is subjected to two simple harmonic motions, one along the X-axis and the other on a line making an angle of 45° with the X-axis. The two motions are given by x = x0 sint and s = s0 sin t Find the amplitude of the resultant motion. ,d d.k ij nks ljy vkorZxfr;k¡ vkjksfir dh tkrh gS] ,d X-v{k ds vuqfn'k rFkk nwljh X-v{k ls 45° dks.k cukus
okyh js[kk ds vuqfn'kA nksuksa xfr;k¡ x = x0 sint
and
s = s0 sin t
}kjk O;Dr dh tkrh gSA ifj.kkeh xfr dk vk;ke Kkr dhft;sA 58.
x
2 0
s02 2 x 0 s0
1/ 2
manishkumarphysics.in
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