TARGET JEE (MAIN/ADVANCED) PRACTICE
CatalyseR atalyseR
PHYSICS
SHEET # 03
www.catalyser.in
CHAPTER: UNIT DIMENSIONS & MEASUREMENT TOPIC: UNITS & DIMENSIONS DIMENSIONS Objective Questions May have ONE or MORE THAN ONE Answers: 1.
2.
Force F and density d are related as
F =
α β +
d
then find the dimensions of α and β.
In the following equations, the distance x is in metres, the time t in seconds and the velocity v in metres/second. What are the SI units of the const ants C 1 and C 2 ? (a)
3.
v
2
x = C1 cos C2 t (c) v = C1e −
= 2 C1x (b)
In the formula p =
nRT V −b
e
−
C 2t
a RTV
. Find the dimensions of a and b where p=pressure, n=no. of moles,
T=temperature, V=volume and R=universal gas constant. 4.
The value of Stefan's constant in CGS system is σ = 5.67 × 10 −5 erg s −1cm −2 K −4 . Its value in SI units is ________.
5.
A particle of mass m is attached to the end of a light string of length
l.
The other end of the string is
attached to a fixed point on a smooth table. The particle is travelling in a horizontal circle on the table with angular velocity ω rad/sec. Assuming that the force in the string depends only upon m, l & ω find an expression for this force. 6.
The displacement of air
particle in
a flute is given by x = a sin (ω t ) cos ( kx ) where ω is a quantity
which depends on the length of the flute (L), gas constant (R), temperature of air (T) and mass per unit mole of the air ( µ ) . Find an expression for ω in terms of all these quantities [Use PV = nRT] 7.
Show that energy, density and power form a fundamental system of units. Find the dimensional formula of universal gravitational constant (G) in this system.
8.
A problem in I. E. Irodov's book is given as below: " A particle of mass m is located in a region where its potential po tential energy [U(x)] depends on the position x as Potential Energy [U(x)] =
a x
2
−
b x
here a & b are positive constants ..."
CatalyseR Eduventures (India) Pvt. Ltd.
1
(i)
Write dimensional formula of a & b
(ii)
If the time period of oscillation which is calculated from above formula is stated by a student as T = 4πa
9.
ma b
2
, check whether his answer is dimensionally co rrect.
Porous rock through which groundwater can move is called an aquifer. The volume V of water that, in time t, moves through a cross section of area A of the aquifer is given by
V / t = KAH / L, where H is the vertical height of the aquifer over the horizontal distance L. This relation is called Darcy's law. The quantity K is the h ydraulic conductivity of the aquifer. What are the SI units of K? 10.
The loss of pressure when a fluid flows through a pipe is given by P = k ρa l V b d c µ where d and l are diameter and length of the pipe respectively, ρ and µ are the mass density and coefficient of viscosity of the fluid, V is the mean velocity of flow through the pipe and k is a numerical constant. Find the values of a, b and c.
11.
If energy, density and power form a fundamental system of units. Find the dimensional formula of momentum in terms of these quantities.
12.
If the velocity of the particle is given b y v=
a
+ bt 2
t The dimensions of a & b are given as _________. 13.
Using 1AU (mean earth-sun distance) =1.5×1011m and parsec as distance at which 1 AU subtends an angle of 1 sec of arc, find parsec in metres.
14.
In two systems of units, the relation between velocity, acceleration and force is given by v2 = =
a1ετ ,
(a)
F2=
m2 m1
F 1 ετ
v1ε 2 τ
, a2
, where ε and τ are constants then find in this new system (b)
L2 L1
END OF PRACTICE SHEET
ANSWER KEY 1
2
CatalyseR Eduventures (India) Pvt. Ltd.
3
2
M
3/ 2
−1
L
2
T
−2
(a) m/s2, (b) m, /sec, (c) m/s, /sec
,
−3/ 2
1/ 2
M , L
[a] = ML5T-2mol-1, [b] = L3
4
5
6
5.67×10−8 J s−1m−1k −4
F = kmω2l
w=
7
k
RT
L
µ
E –2D –1P2
8 (i)
[ a ] = Ux 2 = ML2T −2 L2 = ML4T − 2 ; [b ] = [Ux] = ML2T −2 L = ML3T − 2
(ii) T = 4π a
ma b2
ma 3
= 4π
b2
M M 3 L12T −6 ma 3 Dimenison of RHS = 4π = 2 M 2 L6T −4 b ≠T So, his answer is dimensionally incorrect.
9 V t
=
KAt L
; K=
11 e7/5 ρ1/5 P –3/5
V tA
⇒ K =
L3 t L2
=
12 L ; LT –3
L t
; SI units = m/s
13 9.72
π
10 a = 0; b = 1; c = −2
14
×10 m 16
CatalyseR Eduventures (India) Pvt. Ltd.
(a)
1
ε 2τ 2
(b)
ε 3 τ 3
3
