Dimensions - NEET Physics Questions
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Dimensions

Practice Questions:
Question 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: easy

Which of the following has the dimensions of pressure?

[1994,90]

1. \(MLT^{-2}\)
2. \(ML^{-1}T^{-2}\)
3. \(ML^{-2}T^{-2}\)
4. \(M^{-1}L^{-1}\)
View Answer

Pressure is defined as Force per unit Area. Its dimensions are \(MLT^{-2} / L^2 = ML^{-1}T^{-2}\).

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Question 29: moderate

An equation is given here \[\left(P + \frac{a}{V^2}\right) = b\frac{\theta}{V}\] where P = Pressure, V = Volume and \(\theta =\) Absolute temperature. If (a) and (b) are constants, the dimensions of (a) will be:

1. \(ML^{-5}T^{-1}\)
2. \(ML^5T^{-1}\)
3. \(ML^5T^{-2}\)
4. \(M^{-1}L^5T^2\)
View Answer

From dimensional homogeneity, \([a/V^2] = [P]\). \([a] = [P][V^2]\). Pressure \(P = [ML^{-1}T^{-2}]\), Volume \(V = [L^3]\). So, \([a] = [ML^{-1}T^{-2}][L^3]^2 = [ML^{-1}T^{-2}L^6] = [ML^5T^{-2}]\).

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Question 30: moderate

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]

1. \(ML^2T^{-2}I^{-2}\)
2. \(ML^2T^{-1}I^{-1}\)
3. \(ML^2T^{-3}I^{-2}\)
4. \(ML^2T^{-3}I^{-1}\)
View Answer

Resistance \(R = V/I = W/(QI)\). (W) is work done, (Q) is charge. \(W = [ML^2T^{-2}]\), \(Q = [IT]\). So, \(R = [ML^2T^{-2}] / ([IT][I]) = [ML^2T^{-3}I^{-2}]\).

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