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QUESTION: 1

The dimensional formula for angular momentum is [1988]

Solution:

[Angular momentum ] = [Momentum of inertia] × [Angular velocity] = ML^{2} × T^{–1}

= ML^{2}T^{–1}

OR

In **three dimensions**, the **angular momentum** for a point particle is a pseudovector r × p, the cross product of the particle's position vector r (relative to some origin) and its **momentum** vector; the latter is p = mv in Newtonian mechanics.

In SI base units: kg m^{2} s^{−1}

Dimension: M L^{2}T^{−1}

Derivations from other quantities: L = Iω = r × p

QUESTION: 2

If C and R denote capacitance and resistance, the dimensional formula of CR is [1988]

Solution:

CR = [M ^{-1} L^{-2} T 4 A ^{2} ][M^{1} L^{2} T ^{-3} A^{ -2} ] = [T] = [M^{0} L^{0} T^{1}]

QUESTION: 3

The dimensional formula of torque is [1989]

Solution:

ζ = [Force × distance] = [MLT^{–2}] [L] = ML^{2}T^{–2}

QUESTION: 4

Dimensional formula of self inductance is [1989]

Solution:

Induced emf

where L is the self inductance and is the rate of change of current.

∴ Dimensional formula of L =

QUESTION: 5

Of the following quantities, which one has dimension different from the remaining three? [1989]

Solution:

For angular momentum, the dimensional formula is ML^{2}T^{–1}. For other three, it is ML^{-1}T^{–2}.

QUESTION: 6

If x = at + bt^{2}, where x is the distance travelled by the body in kilometers while t is the time in seconds, then the units of b is [1989]

Solution:

QUESTION: 7

According to Newton, the viscous force acting between liquid layers of area A and velocity gradient ΔV/ΔZ is given by where η is constant called coefficient of viscosity. The dimensional formula of η is [1990]

Solution:

Substitute the dimension al formula of F, A, ΔV and ΔZ on both sides and find that for η.

QUESTION: 8

The frequency of vibration f of a mass m suspended from a spring of spring constant k is given by a relation of the type f = c m^{x} k^{y}, where c is a dimensionless constant. The values of x and y are

Solution:

f = c m^{x} k^{y}; Spring constant k = force/length. [M^{0}L^{0}T^{–1}] = [M^{x }(MT^{–2})^{y}] =[ M^{x} + y T^{–2y}]

⇒ x+y = 0, -2y = -1 or

Therefore, x = -

QUESTION: 9

The dimensional formula of pressure is [1990]

Solution:

[Pressure] = [Force] / [Area]

= ML^{–1}T^{–2}

QUESTION: 10

The dimension al formula for permeability µ is given by [1991]

Solution:

[n] = L^{–1}, [I] = A

QUESTION: 11

A certain body weighs 22.42 gm and has a measured volume of 4.7 cc. The possible error in the measurement of mass and volume are 0.01 gm and 0.1 cc. Then maximum error in the den sity will be [1991]

Solution:

D = M/V

=2 %

QUESTION: 12

P represents radiation pressure, crepresents speed of light and S represents radiation energy striking unit area per sec. The non zero integers x, y, z such that P^{x} S^{y} c^{z} is dimensionless are [1992]

Solution:

Try out the given alter natives.

When x = 1, y = –1, z = 1

= [M^{0}L^{0}T^{0}]

QUESTION: 13

The time dependence of a physical quantity p is given by p = p_{0} exp (– αt^{2}), where α is a constant and t is the time. The constant α [1993]

Solution:

In p = p_{0} exp (–αt^{2}), αt^{2} dimension less

QUESTION: 14

urpentine oil is flowing through a tube of length l and radius r. The pressure difference between the two ends of the tube is P. The viscosity of oil is given by where v is the velocity of oil at a distance a from the axis of the tube. The dimensions of η are

Solution:

QUESTION: 15

In a vernier calliper N divisions of vernier scale coincides with (N – 1) divisions of main scale (in which length of one division is 1 mm). The least count of the instrument should be[1994]

Solution:

Least count = 1MSD – 1 VSD

(∵ N VSD = (N – 1)MSD

QUESTION: 16

In a particular system, the unit of length, mass and time are chosen to be 10 cm, 10 g and 0.1 s respectively. The unit of force in this system will be equivalen t to [1994]

Solution:

[F] = MLT^{–2} = (10g) (10 cm) (0.1s)^{–2}

= (10^{–2} kg) (10^{–1}m) (10^{–1}s)^{–2} = 10^{–1}N.

QUESTION: 17

Which of the following is a dimensional constant? [1995]

Solution:

A quantity which has dimensions and a constant value is called dimensional constant. Therefore, gravitational constant (G) is a dimensional constant.

QUESTION: 18

The percentage errors in the measurement of mass and speed are 2% and 3% respectively.The error in kinetic energy obtained by measuring mass and speed will be [1995]

Solution:

Percentage error in mass

percentage error in speed

Kinetic energy

∴ Error in measurement of kinetic energy

By Binomial Function ,

Reqd. error

∴ %age error = 8%.

QUESTION: 19

An equation is given as :

where P = Pressure, V = Volume & θ = Absolute temperature. If a and b are constants, then dimensions of a will be [1996]

Solution:

According to the principle of homogeinity quantity with same dimension can be added or subtracted.

Hence, Dimension of P = Dimension of

⇒ Dimension of = Dimension of

a = [M L^{5} T^{–2}]

QUESTION: 20

The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and length are 4% and 3% respectively, the maximum error in the measurement of den sity will be [1996]

Solution:

Density =

% error in density = % error in Mass + 3 (% error in length]

= 4 + 3(3) = 13%

QUESTION: 21

Which of the following will have the dimensions of time [1996]

Solution:

QUESTION: 22

The force F on a sphere of radius a moving in a medium with velocity v is given by F = 6πηav.The dimensions of η are [1997]

Solution:

F= 6πη av

= ML^{–1}T^{–1}

QUESTION: 23

The dimensional formula for magnetic flux is [1999]

Solution:

Dimension of magnetic flux = Dimension of voltage × Dimension of time = [ML^{2}T^{–3}A^{–1}] [T] = [ML^{2}T^{–2}A^{–1}]

∵ Voltage =

QUESTION: 24

Which one of the following groups have quantities that do not have the same dimensions? [2000]

Solution:

Force has dimension [MLT^{–2}] while impulse has dimension [MLT^{–1}], both have different dimensions.

QUESTION: 25

The dimensions of Planck’s constant are same as [2001]

Solution:

We know that E = hv

Angular momentum = Iω

= [ML^{2}][T^{–1}] = [ML^{2}T^{–1}]

QUESTION: 26

The unit of the Stefan-Boltzmann's constant is [2002]

Solution:

E = σAT^{4}

E is energy dissipated per second.

QUESTION: 27

The unit of permittivity of free space, ε_{o} is [2004]

Solution:

⇒ unit of ε_{o} is (coulomb)^{2}/ newton -metre^{2}

QUESTION: 28

The dimensions of universal gravitational constant are [1992, 2004]

Solution:

∴ dimension of G is

= M^{–1}L^{3}T^{–2}

QUESTION: 29

The ratio of the dimension of Planck’s constant and that of the moment of inertia has the dimension of [2005]

Solution:

QUESTION: 30

The velocity v of a particle at time t is given by where a, b and c are constant.The dimensions of a, b and c are respectively [2006]

Solution:

Dimension of a. t = dimension of velocity a . t = LT^{-1} ⇒ a = LT^{-2 }

Dimension of c = dimension of t (two physical quantity of same dimension can only be added)

So, dimension of c = T

Dimension of = Dimension of v

So, answer is LT^{–2}, L & T

QUESTION: 31

Dimensions of resistance in an electrical circuit, in terms of dimension of mass M, of length L, of time T and of current I, would be [2007]

Solution:

Dimensions of Resistance,

QUESTION: 32

If the error in the measurement of radius of a sphere is 2%, then the error in the determination of volume of the sphere will be: [2008]

Solution:

Error in the measurement of radius of a sphere = 2%

Volume of the sphere =

∴ Error in the volume =

= 3 × 2% = 6%

QUESTION: 33

Which two of the following five physical parameters have the same dimensions?

(A) Energy density [2008]

(B) Refractive index

(C) Dielectric constant

(D) Young’s modulus

(E) Magnetic field

Solution:

[Energy density] =

[Young’s Modulus] =

QUESTION: 34

If the dimensions of a physical quantity are given by M^{a} L^{b} T^{c}, then the physical quantity will be:

Solution:

QUESTION: 35

The dimension of where ε_{0} is permittivity of free space and E is electric field, is:[2010]

Solution:

represents energy density i.e., energy per unit volume.

QUESTION: 36

The dimensions of are [2011]

Solution:

, so, dimensions are [LTT^{–1}].

QUESTION: 37

The density of material in CGS system of units is 4g/cm^{3}. In a system of units in which unit of length is 10 cm and unit of mass is 100 g, the value of density of material will be [2011M]

Solution:

In CGS system,

The unit of mass is 100g and unit of length is 10 cm, so

density =

= 40 unit

QUESTION: 38

The dimensions of (μ_{0}ε_{0})^{–1/2 }are : [2012M]

Solution:

: speed of light

where ε_{0} = permittivity of free space

μ_{0} = permeability of free space

So dimension LT^{–1}

QUESTION: 39

In an experiment four quantities a, b, c and d are measured with percentage error 1%, 2%, 3% and 4% respectively. Quantity P is calculated as follows error in P is [NEET 2013]

Solution:

100%.

= 3 × 1% + 2 × 2% + 3% + 4% = 14%

QUESTION: 40

The pair of quantities having same dimensions is [NEET Kar. 2013]

Solution:

Work = Force × displacement Torque = Force × force arm

= mass × acceleration × length

= [M] × [LT^{–2}] × [L] = [M L^{2}T^{–2}]

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