A stone is dropped from a height of 19.6m, above the ground while a second stone is simultaneously projected from the ground with sufficient velocity to enable it to ascend 19.6m. When and where the stones would meet? At any time t, s1 s2 19.6 s1
1
gt 2 ; s2 v0 t
2 v0 t 19.6
1 2
gt 2
From stone 2, v 0 v02 2 9. 9 .81 19.6 v0 19.6 m/s t 1 s s s2 19.6 1
1 2
9.81 1
2
s 14.7 m The relation 3t =√(3x) + 6 describes the displacement of a particle in one direction, where x is in metres
and t in seconds. Find the displacement when the velocity is zero. x v
3t 6 3 dx
dt t 2 x
2
2 3t 6 0
2 2 6
2
0 3 Cars A and B are travelling in adjacent lanes along a straight road. At time, t = 0 their positions and speeds are as shown in the diagram. If car A has a constant acceleration of 0.6m/s2 and car B has a constant deceleration of 0.46m/s2, determine when when A will overtake B.
s B s A 30 v0 B t
1 2
aB t 2 30 v0 At
1 2
a At 2
20t 0.5 0.46 t 2 30 13t 0.5 0. 0 .6 t 2 t 0.9 s
A stone is dropped into the water from a tower 44.1m above the ground. Another stone is thrown vertically down 1.0 s after the first one is dropped. Both the stones strike the ground at the same time. What was the initial velocity of the second stone? 1 s1 gt 12 2 44.1 0.5 9.81 t 2 t 1 2.998 s t2 t 1 1 1.998 s s2 s1 v02 t2
1 2
gt 22
44.1 v02 1.998 0.5 9.81 1.998 2 v02 12.25 m/s
A balloon is ascending at the rate of 9.8 m/s at a height of 98m above the ground when a packet is dropped. How long does it take the packet to reach the ground? v0 9.8 m/s 2 s 98 m s v0 t
1 2
98 9.8t
gt 2 1 2
9.81 t 2
t 5.58 s
A container of mass 200 kg rests on the back of an open truck. If the truck accelerates at 1.5m/s2, what is the minimum coefficient of static friction between the container and the bed of the truck required to prevent the container from sliding off the back of the truck. F f m a m g
a g
1.5 9.81
0.153
A thin cylindrical wheel of radius r = 40 cm is allowed to spin on a frictionless axle. The wheel, which is initially at rest, has a force applied of magnitude 50N at an angle of 20◦ to the tangent as shown. The wheel has a moment of inertia equal to 20 kgm2.
Find the angular acceleration of the wheel By Newton's 2nd law for rotation:
T I T F r 50 cos 20 0.4 T 18.79 Nm 18.79 20 0.94 rad/s 2
In the last second of a free fall, a body covered three-fourth of its total path: (a) For what time did the body fall? (b) From what height did the body fall? 1 h gt 2 2 h 1 2 g t 1 4 2 1 t 2 1/ 4 t 12 t 2 s h
1 2
9.81 2
2
h 19.6 m
Two objects are projected horizontally in opposite directions from the top of a tower with velocities u1 and u2. Find the time when the velocity vectors are perpendicular to each other and the distance of separation at that instant.
v1 u1i gtj v2 u2 i gtj For perpendicular vectors, v1 v2 0
`
u1i gtj u2 i gtj 0 2
u1u2 gt 0 t
u1u2 g
A body slides down an incline which has coefficient of friction μ = 0.5. Find the angle θ if the incline of
the normal reaction is twice the resultant downward force along the incline. N 2 F x N Normal force mg cos Fx - resultant downward force Fx mg sin mg cos mg cos 2 mg sin mg cos mg cos 1 2 2 mg sin tan
1 2 2
1 2 0.5 2
45
A ramp in an amusement park is frictionless. A smooth object slides down the ramp and comes down through a height h. What distance d is necessary to stop the object on the flat track if the coefficient of friction is μ.
PE W f mgh fd mg h mg d d h /
From an elevated point A, a stone is projected vertically upward. When the stone reaches a distance h below A, its velocity is double of what it was at a height h above A. What is the greatest height obtained by the stone above A? Combining (1), (2) and (3), From the problem, v2 2v1
(1)
v02
10
gh 3 When the stone is h above A, At the maximum height H, v 0; v12 v02 2 gh (2) 0 v02 2 gH When the stone is h below A, v02 10 gh 1 5h 2 2 v2 v0 2 gh (3) H 2 g 3 2 g 3
A particle moves according to the law x = A sin πt, where x is the displacement and t is time. Find the distance traversed by the particle in 3.0 s. x A sin t A sin t
2
; T 2 T In 0.5s, the distance covered is A.
In 3s, the distance covered is 6A . A particle projected up passes the same height h at 2 and 10 s. Find h if g = 9.8m/s2. ht 2 ht 10 v0 t2 2v0
1 2 1
gt22 v0 t10 2
1 2
gt102 1
2
9.81 2 10v0 9.8110
2 v0 58.86 m/s
h 58.86 2
2
1 2
2
9.81 2
h 98 m
Water drips from the nozzle of a shower onto the floor 2.45mbelow. The drops fall at regular interval of time, the first drop striking the floor at the instant the third drop begins to fall. Locate the second drop when the first drop strikes the floor.
Time taken for the first drop to reach the floor is 2h
t1
g
2 2.45 9.8
1 2
s
As the time interval between the first and second drop is equal to that of the second and the third drop (drops dripping at regular intervals), time taken by the second drop is t2
1 2 2
s
1 S gt 9.8 2 2 2 2 1
2 2
1
2
S 0.6125 m
The velocity –time graph for the vertical component of the velocity of an object thrown upward from the ground which reaches the roof of a building and returns to the ground is shown. Calculate the height of the building.
MODE 3 2 (Linear Reg.) x
y
0
30
4
10
A STO A B STO B
MODE 1 t
A Bt dt h A Bx dx h
0
4
0
h 40 m
A ball is dropped into a lake from a diving board 4.9m above the water. Ithits the water with velocity v and then sinks to the bottom with the constant velocity v. It reaches the bo ttom of the lake 5.0 s after it is dropped. Find (a) the depth of the lake and (b) the average velocity of the ball.
(a) Time for the ball to reach water 2h
t1
g
2 4.9
9.8
1.0s
v gt 1 9.8 1.0 9.8 m/s Time taken to reach the bottom of the lake from surface t2 5.0 1.0 4.0 s
(b) vave
displacement time
4.9 39.2 5
vave 8.82 m/s
d vt 2 9.8 4 39.2 m An elevator of height h ascends with constant acceleration a. When it crosses a platform, it has acquired a velocity u. At this instant a bolt drops from the top of the elevator. Find the time for the bolt to hit the floor of the elevator. For the motion of the bolt, h v0 t
1
a ' t 2 ; v0 0
2 The acceleration of the bolt relative to the elevator is a' g a h
1 2
t
g a t 2 2h g a
A particle moving in the xy-plane has velocity components dx/dt = 6 + 2t and dy/dt = 4 + t. Find the magnitude of the acceleration and its direction with respect to the x-axis. dy dx Pol(2,1) v y 4 t v x 6 2t dt dt r a 2.236 m/s 2 dv y dv a x x 2 1 a y 26.565 dt dt From the ground an object is projected upward with sufficient velocity so that it crosses the top of a tower in time t1 and reaches the maximum height. It then comes down and recrosses the top of the tower in time t2, time being measured from the instant the object was projected up. A second object released from the top of the tower reaches the ground in time t3. Find t3.
For the motion of the object, Let the two roots be t and t . In the quadratic equation 1 2 2 1 2 ax bx c 0, c / a is the product of the roots. s v0 t gt 2 c h 2h t1t 2 1 2 1 h v0 t gt a g g 2 2 1 2 But t3 is the total time of flight gt v0 t h 0 2 2h t1t 2 t3 g A shell is fired at an angle θ with the horizontal up a plane inclined at an angle α. What should be the value of θ to obtain the maximum range?
For maximum range Rmax
2
4
A stone is thrown from ground level over horizontal ground. It just clears three walls, the successive distances between them being r and 2r . The inner wall is 15/7 times as high as the outer walls which are equal in height. The total horizontal range is nr, where n is an integer. Find n. n 4
A projectile is fired at an angle of 60° to the horizontal with an initial velocity of 800 m/s. Determine the equation for its parabolic path. y x tan
gx 2 2v02 cos 2
y x tan 60
9.81 x 2 2 800 2 cos 60
2
y 1.732 x 3.0656 10 5 x 2
An object is projected horizontally with velocity 10 m/s. Find the radius of curvature of its trajectory in 3 s after the motion has begun.
1 y '
2
d
1.5
y ''
y x tan
gx 2 2v02 cos 2
x tan 0 x30 STO A y ' 2 2 dx 2 10 cos 0 d 9.81x 2 x tan 0 x 3 00. 001 STO B 2 2 dx 2 10 cos 0
x v0 cos t 10 cos 0 3 30
1 A2
9.81x 2
1.5
B A
305 m
0.001
Three vector forces F1, F2 and F3 act on a particle of mass m = 3.80 kg as shown:
(i) Calculate the magnitude and direction of the net force acting on the particle. R 600 80145 40 45 R 28.7637.7 (ii) Calculate the particle’s acceleration.
F ma 28.76 3.8a a 7.57 m/s 2
(iii) If an additional stabilizing force F4 is applied to create an equilibrium condition with a resultant net force of zero, what would be the magnitude and direction of F4? F4 R F 4 28.76 142.3
A thin cylindrical wheel of radius r = 40 cm is allowed to spin on a frictionless axle. The wheel, which is initially at rest, has a tangential force applied at right angles to its radius of magnitude 50N as shown. The wheel has a moment of inertia equal to 20 kgm2.
Calculate (i) The torque applied to the wheel T Fr 50 40 102 T 20 Nm
(ii) The angular acceleration of the wheel T I 20 20 1 rad/s
(iii) The angular velocity of the wheel after 3 s 0 t 0 13 3 rad/s
(iv) The total angle swept out in this time 1 0 t t 2 2 1 2 0 1 3 2 4.5 rad
A thin cylindrical wheel of radius r = 40 cm is allowed to spin on a frictionless axle. The wheel, which is initially at rest, has a force applied of magnitude 50N at an angle of 20◦ to the tangent as shown. The wheel has a moment of inertia equal to 20 kgm2.
Find the torque applied to the wheel T Fr 50 40 102 cos 20 T 18.8 Nm
A wheel of radius r and weight W is to be raised over an obstacle of height h by a horizontal force F applied to the centre. Find the minimum value of F.
T
D
0
F CD W BD F W
F
BD CD
OD 2 OB2 CE DE
r2 r h r h
2
h 2r h r h
A ladder leaning against a smooth wall makes an angle α with the horizontal when in a position of limiting equilibrium. Show that the coefficient of friction between the ladder and the ground is 0 .5cot α.
F 0 y
N1 mg
F 0 x
F N 2 mg
T
ground
0
N 2 l sin mg 0.5 l cos mg sin 0.5 mg cos 0.5cot
Three blocks of mass m1, m2 and m3 interconnected by cords are pulled by a constant force F on a frictionless horizontal table. Find
(a) Common acceleration ‘a’
a
F m1 m2 m3
(b) Tensions T1 and T2 T1 m1a T2 m1 m2 a
A block of mass m1 sits on a block of mass m2, which rests on a smooth table. If the coefficient of friction between the blocks is μ, find the maximum force that can be applied to m2 so that m1 may not
slide.
F ma Fmax m1 m2 a At m1, the applied force is friction. f m1 a m1 g a g Fmax m1 m2 g
Two blocks m1 and m2 are in contact on a frictionless table. A horizontal force F is applied to the block m1. (a) Find the force of contact between the blocks.
F12 F
m2 m1 m2
(b) Find the force of contact between the blocks if the same force is applied to m2 rather than to m1.
F12 F
m1 m1 m2
A block is placed on a ramp of parabolic shape given by the equation y = x2/20. If μs = 0.5, what is the
maximum height above the ground at which the block can be placed without slipping? tan y ' s xmax /10 0.5 xmax 5 2 ymax xmax / 20 52 / 20
ymax 1.25 m
A wedge of mass M is placed on a horizontal floor. Another mass m is placed on the incline of the wedge. Assume that all surfaces are frictionless, and the incline makes an angle θ with the horizontal.
The mass m is released from rest on mass M, which is also initially at rest. Find the accelerations of M and m.
Two masses in an Atwood machine are 1.9 and 2.1 kg, the vertical distance of the heavier body being 20 cm above the lighter one. After what time would the lighter body be above the heavier one by the same vertical distance? Neglect the mass of the pulley and the cord. m m2 a 1 g (m1 m2 ) m1 m2 a
2.1 1.9 2.1 1.9
9.81
a 0.49 m/s 2
A body takes 4/3 times as much time to slide down a rough inclined plane as it takes to slide down an identical but smooth inclined plane. Find the coefficient of friction if the angle of incline is 45◦. t1
2 s
a1 g sin g cos
4
a2
For smooth incline
3
ma2 mg sin
7/16
For rough incline, ma1 mg sin mg cos
a2 g sin
a1
; t 2
a1
2s a2
;
t 1 t 2
4 3
g sin 45 g sin 45 g cos 45
The constant forces F1 = ˆi +2 ˆj +3ˆ k N and F2 = 4ˆi −5 ˆj −2ˆk N act together on a particle during a displacement from position r2 = 7ˆ k cm to position r1 = 20ˆi + 15 ˆj cm. Determine the total work done
on the particle. F F1 F2 5i 3 j k r12 r1 r2 0.2i 0.15 j 0.07.k m W F r 12 W 5 0.2 3 0.15 1 0.07 W 0.48 J A body slides down a rough plane inclined to the horizontal at 30◦. If 70% of the initial potential energy
is dissipated during the descent, find the coefficient of sliding friction.
PE W f 0.7 mgh fs h sin f N mg cos 0.7 mg s sin30 mgs cos30 0.404
Consider an elastic collision between an incident particle of mass m with M initially at rest (m>M). What is the largest possible scattering angle? max = sin -1 (M/m)
A machine gun fires 100 g bullets at a speed of 1000 m/s. The gunman holding the machine gun in his hands can exert an average force of 150N against the gun. Find the maximum number of bullets that can be fired per minute. p F mvn ' t F 150 1.5 / s n' mv 0.1 1000 n 1.5 / s 60 s 90
A wheel rotates with a constant angular acceleration α = 3 rad/s2. At time t = 1.0 s after the motion begins the total acceleration of the wheel bec omes a = 12√10 cm/s2. Determine the radius of the wheel.
aT
2 at2 arad
aT
r r 2
2
2
0 t 0 3 1 3 12 10 10 2
2
3r 9 r
2
r 4 cm
A car travels around a horizontal bend of radius R at constant speed V. If the road surface has a coefficient of friction μs, what is the maximum speed, Vmax, at which the car can travel without sliding? Given μs = 0.85 and R = 150 m, what is Vmax?
Vmax
gr
V max
0.85 9.81 150
V max 35.36 m/s
A coin is placed at a distance r from the centre of a gramophone record rotating with angular frequency ω = 2π f . Find the maximum frequency for which the coin will not slip if μ is the co efficient of friction.
FC f mr 2 mg g
f
r
2 f
1
g
2
r
A curve on a highway has a radius of curvature r . The curv ed road is banked at θ with the horizontal. If the coefficient of static friction is μ, obtain an expression for the maximum speed v with which a car can go over the curve without skidding. Find v if r = 100 m, θ = 30◦, g = 9.8m/s2, μ = 0.25. v
gr tan ;
v 9.8 100 0.25 tan 30 v 28.47 m/s
A car travels at a constant speed of 14.0m/s round a level circular bend of radius 45 m. What is the minimum coefficient of static friction between the tyres and the road in order for the car to go round the bend without skidding? vmax
gr
14
9.81 45
0.444
A solid cylinder of mass m and radius R rolls down an inclined plane of height h without slipping. Find the speed of its centre of mass when the cylinder reaches the bottom. For solid cylinders on an incline, v
4 3
gh
Calculate the gravitational force between two lead spheres of radius 10 cm in contact with one another, G = 6.67 × 10−11 MKS units. Density of lead = 11,300 kg/m3.
F G
m1m 2 r2
G
m2
r 2 4
0.13 3
m V 11300 m 47.33 kg r 0.2 m F G
47.332 0.2
2
3.74 106 N
The length of the longest string in a piano is 2.0m and the wave velocity of the string is 120 m/s. Find the frequency of thethirdharmonic. v 120 30 Hz f 1 2 L 2 2 f3 3 f 1 3 30 90 Hz
A piano string of length 0.6m is under a tension of 300N and vibrates with a fundamental frequency of 660 Hz. What is the mass density of the string? f 1
1
T
2 L
660
1
300
2 0.6
4.78 10 4 kg/m
A wave of f requency 250 cycles/s has a phase velocity 375 m/s. (a) How far apart are two points 60◦ out of phase? What is the phase difference between two displacements at a certain point at time 10−3 s
apart? 2 ft 60
2 250t 180 t 666 s
2 ft 2 250 10 3
x vt 375 666 10 6
x 0.25 m
2
rad 90
A small source of sound radiates energy uniformly at a rate of 4W. Calculate the intensity and the intensity level at a point 25 m from the source if there is no absorption. P 4 I 4 R 2 4 252 I 5.09 104 W/m 2 I 10 log
I I 0
10log
5.09 104 1012
I 87 dB Find the theoretical speed of sound in hydrogen at 0◦C. For a diatomic gas γ = 1.4 and for hydrogen M =
2.016 g/mol; the universal gas constant R = 8.317 J/mol/K. v v
RT M 1.4 8.317 273 2.016 103
v 1256 m/s
Two sound waves have intensities 0.4 and 10W/m2, respectively. How many decibels is one louder than the other? 10 dB 10 log 14 dB 0.4 If one sound is 6.0 dB higher than another, what is the ratio of their intensities? R 106/10 3.98 Two trains move towards each other at a speed of 90 km/h relative to the earth’s surface. One gives a
520 Hz signal. Find the frequency heard by the observer on the other train. v v L f f 0 v vS v L vS 90 kph 3.6 25 m/s f 520
340 25 340 25
f 600 Hz Two trains move away from each other at a speed of 25 m/s relative to the earth’s surface. One gives a
520 Hz signal. Find the frequency heard by the observer on the other train (sound velocity = 330m/s). v v L f f 0 v vS v L vS 90 kph 3.6 25 m/s f 520
330 25 330 25
f 447 Hz
Two whistles are sounded with frequencies of 548 and 552 cycles/s, respectively. A man directly in the line between them walks towards the lower pitched whistle at 1.5m/s. Find the beat frequency that he hears. Assume thesound velocity of 330 m/s. fb f1 f 2 Listener moves away from higher freq., f1 f 1 '
v vS
f 2 f 2 '
v v v s v
552 548
330 1.5 330 330 1.5 330
549.5 Hz 550.5 Hz
f b 550.5 549.5 1 Hz
The radius of a water pipe decreases from 10 to 5 cm. If the average velocity in the wider portion is 4 m/s, find the average velocity in the narrower region.
Q A1v1 A2 v2 2
2
10 4 5 v2 v2 16 m/s A garden sprinkler has 80 small holes each 2.5mm2 in area. If water is supplied at the rate of 2 × 10−3
m3/s, find the average velocity of the spray. Q
QT
n Q Av
2 10 3 80
2.5 105
2.5 10 5 2.5 10 6 v v 10 m/s
If the speed of flow past the lower surface of the wing of an aeroplane is 100 m/s, what speed of flow over the upper surface would give a pressure difference of 1000 Pa? Assume an air density of 1.293 kg/m3. Using Bernoulli's Equation. 1 1 p1 v12 gh1 p2 v22 gh2 2 2 1 p p2 p1 v12 v22 2 1 1000 1.293 v 2 1002 2 v 107.46 m/s
A venturi meter has a pipe diameter of 4 cm and a throat diameter of 2 cm. The velocity of water in the pipe section is 10 cm/s. Find (a) the velocity in the throat; (b) the pressure drop. 1 1 p1 v12 gh1 p2 v22 gh2 2 2 Q A1v1 A2 v2 1 p p2 p1 v12 v22 2 2 2 4 10 2 vthroT 1 p 1000 40 2 10 2 v 40 m/s 2
p 750 kPa A bubble of gas rises from the bottom of a lake 30m deep. At what depth will the volume be thrice as great as it was originally?
p1V1 p2V 2 V2 V1 3
3
p1 p2
101325 1000 9.81 30 101325 1000 9.81 h
h 3.11 m A spherical liquid drop has a diameter of 2mm and is given a charge of 2 × 10−15 C.
(i) What is the potential at the surface of the drop? V
Q 4 0 r
2 10 15 4 0 1 10 3
V 0.018 V
(ii) If two such drops coalesce to form a single drop, what is the potential at the surface of the drop so formed? QT 2Q 2 10 5
4 (10 3 ) 3 3
VT 2V 2
VT 8.3776 10 9
4 3
RT 3
RT 1.26 mm V T
2 10 `15 4 0 1.26 10 3
V T 0.0286 V
A spark is produced between two insulated surfaces, maintained at a constant difference of 5 × 106 V. If the energy output is 10−5 J, calculate the charge transferred. How many electrons have flowed? W QV Q e
105 5 10
6
2 10 12 C
2 1012
1.248 10 7 e
CONST 23 If the breakdown field strength of air is 5×106 V/m how much charge can be placed on a sphere of radius 1 mm? Q V 4 0 r Q 5 106 4 0 (10 3 ) Q 5.5 10 7 C
What is the maximum charge that can be given to a sphere of diameter 10 cm if the breakdown voltage of air is 2 × 104 V/cm. V
Q 4 0 r Q
2 104 102
4 0 5 102
5
Q 1.11 10
C
What is the capacitance of a conducting sphere of radius a? Q C V Q V 4 0 a Q Q
C
4 0 a
4 0 a
Two capacitors C1 and C2 are connected in parallel a nd their combined capacitance is measured as 9μF. When they are combined in series their capacitance is 2μF. What are the individual capacitances?
C1 C 2 9 C1C2 C1 C 2
2
C1C2 9
C1C 2 18 MODE 5-3
1
-9 18
C1 6 F; C 2 3 F
Two parallel plates, each of area 1m2, are separated by a distance 0.001m and have a capacity of 0.1μF. What must then the dielectric constant of the material separating the plates be? A C d 1 0.1 10 6 r 0 0.001 r 11.3 A capacitor of capacitance 5μF is charged up to a PD of 250V. Its terminals are then connected to those
of an uncharged capacitor of capacitance 20μF. What would be the resulting voltage?
QT C1V 1 5 106 250 1.25 mC V T
QT C T
1.25 10 3
5 20 10 6
50 V
Two spherical metallic shells of radii a and b (b > a) constitute a capacitor with the outer shell grounded and contact is made with the inner one through a hole in the outer one. What is the capacitance? 4 0 ab C b a Find the radius of a circular orbit of an electron of energy 5 keV in a fi eld of 10−2 T. mv 2
qvB
r r
mv
K
qB 1 2
mv2 1
5 10 CONST 23 2 CONST 03 v 3
2
v 4.194 10 7 m/s
CONST 03 4.194 10 7 r CONST 23 102 r 2.38 cm
An electric field of 1500 V/m and a magnetic field act on an electron moving with a speed of 3000 m/s. If the resultant field is to be zero what should be the strength of the magnetic field (in Wb/m2). Felectric F magnetic q E q vB 1500 3000 B B 0.5 T An electron moves in a circle of radius 1.9m in a magnetic field of 3 × 10−5 T. Calculate (a) the speed of
electrons and (b) time taken to move round the circle.
mv 2
qvB
r
v
qBr
v
qB r m
r
107 1.9
m
CONST 23 3 105 1.9 5.28 106 2 v T CONST 03 T 1.19 s
v 10 m/s 7
In the Bohr model of hydrogen atom the electron revolves in a circular orbit of radius 0.53Å with a time period of 1.5 × 10−16 s. Find the corresponding current.
i
q t
CONST 23 1.5 1016
1.07 mA
Two parallel wires 20 cm apart attract each other with a force of 10−5 N/m length. If the current in one
wire is 10 A, find the magnitude of current in the other wire? F I1 I 2 L 2 r 10 I 2 105 2 20 10 2 I 2 1.0 A
Calculate the energy density at the centre of a circular loop of wire 10 cm radius carrying a current of 100 A. u B
B 2 2 0 0 I 2r
0 100 2 10 10 2
B 6.283 10 4 T u
6.283 104
2
2 0
u 0.157 J/m 3
A metal disc of radius 0.1m spins about a horizontal axis lying in the magnetic meridian at a speed of 5 rev/s. If the horizontal component of the earth’s field is B = 2×10−5 Wb/m2, calculate the potential difference between the centre and the outer edge of the disc.
V NBA V 2 5 1 2 10 5 0.12 V 1.97 10 5 V
