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The fundamental physical quantities that have same dimensions in the dimensional formulae of torque and angular momentum are

Mass, time

Time, length

Mass, length

Time, mole

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Explanation

Torque = $[M L^{2} T^{- 2}]$ , Angular momentum = $[M L^{2} T^{- 1}]$

So mass and length have the same dimensions.

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If pressure P, velocity V and time T are taken as fundamental physical quantities, the dimensional formula of force is

$P V^{2} T^{2}$

$P^{- 1} V^{2} T^{- 2}$

$P V T^{2}$

$P^{- 1} V T^{2}$

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Explanation

Let $F \propto P^{x} V^{y} T^{z}$

by substituting the following dimensions :

$[P] = [M L^{- 1} T^{- 2}]$ , $[V] = [L T^{- 1}], [T] = [T]$

and comparing the dimension of both sides

$x = 1, y = 2, z = 2$ , so $F = P V^{2} T^{2}$

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The physical quantity which has dimensional formula as that of $\frac{Energy}{Mass \times Length}$ is

Force

Power

Pressure

Acceleration

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Explanation

$\frac{Energy}{mass \times length} = \frac{[M L^{2} T^{- 2}]}{[M] [L]} = [L T^{- 2}]$

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If energy (E), velocity (v) and force (F) be taken as fundamental quantity, then what are the dimensions of mass

$E v^{2}$

$E v^{- 2}$

$F v^{- 1}$

$F v^{- 2}$

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Explanation

Let $m \propto E^{x} v^{y} F^{z}$

By substituting the following dimensions :

$[E] = [M L^{2} T^{- 2}], [v] = [L T^{- 1}], [F] = [M L T^{- 2}]$

and by equating the both sides

$x = 1, y = - 2, z = 0$ . So $[m] = [E v^{- 2}]$

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Dimensions of luminous flux are

$M L^{2} T^{- 2}$

$M L^{2} T^{- 3}$

$M L^{2} T^{- 1}$

$M L T^{- 2}$

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Explanation

$M L^{2} T^{- 3}$

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A physcial quantity x depends on quantities y and z as follows: x = Ay + B tan Cz, where A, B and C are constants. Which of the following do not have the same dimensions

x and B

C and z^–1

y and B/A

x and A

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Explanation

$x = A y + B \tan C z$

From the dimensional homogenity

$[x] = [A y] = [B] \Rightarrow [\frac{x}{A}] = [y] = [\frac{B}{A}]$

$[C z] = [M^{0} L^{0} T^{0}] =$ Dimension less

x and B; C and $Z^{- 1}; y$ and $\frac{B}{A}$ have the same dimension but x and A have the different dimensions.

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Which of the following pair does not have similar dimensions

Stress and pressure

Angle and strain

Tension and surface tension

Planck's constant and angular momentum

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Explanation

Tension = $[M L T^{- 2}]$ , Surface Tension = $[M T^{- 2}]$

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Out of the following which pair of quantities do not have same dimensions

Planck's constant and angular momentum

Work and energy

Pressure and Young's modulus

Torque & moment of inertia

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Explanation

Torque = $[M L^{2} T^{- 2}]$ , Moment of inertia = $[M L^{2}]$

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Identify the pair which has different dimensions

Planck's constant and angular momentum

Impulse and linear momentum

Angular momentum and frequency

Pressure and Young's modulus

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Explanation

Angular momentum = $[M L^{2} T^{- 1}]$ , Frequency = $[T^{- 1}]$

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The dimensional formula $M^{0} L^{2} T^{- 2}$ stands for

Torque

Angular momentum

Latent heat

Coefficient of thermal conductivity

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Explanation

Latent Heat $L = \frac{Q}{m} = \frac{Energy}{mass} = \frac{[M L^{2} T^{- 2}]}{[M]} = [L^{2} T^{- 2}]$

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