Dimensions

Practice Questions:

Question 41: difficult

Planck’s constant \(h\), speed of light in vacuum \(c\), and Newton’s gravitational constant \(G\), are three fundamental constants. Which of the following combinations of these has the dimension of length?

[2016-II]

1. \(\sqrt{\frac{hc}{G}}\)

2. \(\sqrt{\frac{Gc}{h^{3/2}}}\)

3. \(\frac{\sqrt{hG}}{c^{3/2}}\)

4. \(\frac{\sqrt{hG}}{c^{5/2}}\)

View Answer

The Planck length formula is \(l_P = \sqrt{\frac{hG}{c^3}}\). Checking option c: \(\frac{\sqrt{hG}}{c^{3/2}} = h^{1/2}G^{1/2}c^{-3/2}\). \(h=[ML^2T^{-1}]\), \(G=[M^{-1}L^3T^{-2}]\), \(c=[LT^{-1}]\). Thus, \([M^{1/2}L^1T^{-1/2}] [M^{-1/2}L^{3/2}T^{-1}] [L^{-3/2}T^{3/2}] = [M^0L^{1+3/2-3/2}T^{-1/2-1+3/2}] = [L]\).

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Question 42: moderate

Which pair have not equal dimensions?

[2000]

1. Energy and torque

2. Force and impulse

3. Angular momentum and Planck's constant

4. Elastic modulus and pressure

View Answer

Force has dimensions \(MLT^{-2}\) and impulse has dimensions \(MLT^{-1}\). These are not equal.

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Question 43: difficult

If energy \((E)\), velocity \((V)\), and time \((T)\), are chosen as the fundamental quantities, the dimensional formula of surface tension will be:

[2015]

1. \(EV^{-1}T^{-1}\)

2. \(EV^{-2}T^{-2}\)

3. \(E^{-2}V^{-1}T^{-3}\)

4. \(EV^{-2}T^{-1}\)

View Answer

Surface tension \(gamma\) has dimensions \(MT^{-2}\). Given fundamental quantities are energy \(E = [ML^2T^{-2}]\), velocity \(V = [LT^{-1}]\), and time \(T = [T]\). Let \(\gamma = E^x V^y T^z\).

Equating dimensions: \(MT^{-2} = (ML^2T^{-2})^x (LT^{-1})^y (T)^z = M^x L^{2x+y} T^{-2x-y+z}\). Comparing powers: \(x=1\), \(2x+y=0 ⇒ y=-2\), \(-2x-y+z=-2 ⇒ -2(1)-(-2)+z=-2 ⇒s z=-2\). Thus, surface tension dimensions are \(EV^{-2}T^{-2}\).

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Question 44: moderate

The dimensions of impulse are equal to that of:

[1996]

1. Pressure

2. Linear momentum

3. Force

4. Angular momentum

View Answer

Impulse is defined as change in momentum. Therefore, their dimensions are equal, \(MLT^{-1}\).

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Question 45: difficult

If dimension of critical velocity of liquid flowing through a tube are expressed as \(v_c \propto \eta^x \rho^y r^z\) where \(\eta\) and \(\rho\) are the coefficient of viscosity of liquid, density of liquid and radius of the tube respectively, then the values of \(x, y\) and \(z\) are given by:

[2015-Re]

1. 1, 1, 1

2. 1, -1, -1

3. -1, -1, 1

4. -1, -1, -1

View Answer

Critical velocity \(v_c = [LT^{-1}]\). Coefficient of viscosity \(\eta = [ML^{-1}T^{-1}]\). Density \(\rho = [ML^{-3}]\). Radius \(r = [L]\). Assume \(v_c \propto \eta^x \rho^y r^z\). Equating dimensions: \([LT^{-1}] = ([ML^{-1}T^{-1}])^x ([ML^{-3}])^y ([L])^z = [M^{x+y} L^{-x-3y+z} T^{-x}]\). Comparing powers: \(x+y = 0\), \(-x-3y+z = 1\), \(-x = -1\). From \(-x = -1\), we get \(x=1\). From \(x+y = 0\), we get \(1+y = 0 ⇒ y=-1\). From \(-x-3y+z = 1\), we get \(-1-3(-1)+z = 1⇒ -1+3+z=1 ⇒ 2+z=1 ⇒ z=-1\). Therefore, \(x=1, y=-1, z=-1\).

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Question 46: easy

Which of the following dimensions will be the same as that of time?

[1996]

1. \(L/R\)

2. \(C/L\)

3. \(LC\)

4. \(R/L\)

View Answer

The ratio \(L/R\) has the dimensions of time, \(T\).

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Question 47: moderate

If Force \((F)\), Velocity \((V)\), and Time \((T)\), are taken as fundamental units, then the dimensions of mass are:

[2014]

1. \(FVT^{-1}\)

2. \(FVT^{-2}\)

3. \(FV^{-1}T^{-1}\)

4. \(FV^{-1}T\)

View Answer

Given fundamental units: Force \(F = [MLT^{-2}]\), Velocity \(V = [LT^{-1}]\), Time \(T = [T]\). We want to find dimensions of Mass \(M = F^x V^y T^z\). Equating dimensions: \([M] = [MLT^{-2}]^x [LT^{-1}]^y [T]^z = [M^x L^{x+y} T^{-2x-y+z}]\). Comparing powers: \(x=1\), \(x+y=0 ⇒ 1+y=0 ⇒ y=-1\), \(-2x-y+z=0 ⇒ -2(1)-(-1)+z=0⇒ -2+1+z=0 ⇒ z=1\). Thus, mass dimensions are \(FV^{-1}T\).

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Question 48: easy

Which of the following is a dimensional constant?

[1995]

1. Relative density

2. Gravitational constant

3. Refractive index

4. Poisson ratio

View Answer

A dimensional constant is a physical constant with dimensions. The Gravitational constant \(G\) is a dimensional constant with dimensions \(M^{-1}L^3T^{-2}\).

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Question 49: easy

The dimensions of \(RC\) is:

[1995]

1. Square of time

2. Square of inverse time

3. Time

4. Inverse time

View Answer

The product \(RC\) represents the time constant of an \(RC\) circuit, and its dimension is time, \(T\).

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Question 50: easy