Question

Which of the following pairs does not have the same dimensions?
A. impulse and momentum.
B. moment of inertia and moment of force.
C. angular momentum and Planck’s constant.
D. work and torque.

Answer 1

The dimensions of each of the quantities should be found and compared. The one pair with different dimensions will be the answer for the question. The moment of force is the sum of the angular acceleration and the moment of inertia. These details may help you in finding the answer for the question.

Complete answer:
Let us check the dimensional formula of each of the quantities mentioned in the question.
Impulse is the quantity given as the product of the force and the time taken. The dimension of the impulse can be written as,
$\begin{aligned} & \left[ impulse \right]=\left[ force \right]\left[ time \right] \\\ & \left[ impulse \right]=\left[ ML{{T}^{-2}} \right]\left[ T \right]=\left[ ML{{T}^{-1}} \right] \\\ \end{aligned}$
The dimensional formula of the momentum can be calculated by the equation,
$\left[ P \right]=\left[ m \right]\left[ v \right]$
Substitute the dimensional formula of mass and velocity in the equation will give,
$\left[ P \right]=\left[ M \right]\left[ L{{T}^{-1}} \right]=\left[ ML{{T}^{-1}} \right]$
Therefore the first pair has the same dimension.
The moment of force is defined as the product of the moment of forces and the angular acceleration of the body. This can be mathematically written as,
$\tau =I\alpha$
Where $I$ be the moment of inertia, $\alpha$ be the angular acceleration and $\tau$ be the moment of force. Therefore these quantities are having different dimensions.
Angular momentum is shown as the dimensional formula using the equation,
$L=mvr$
Substituting the dimensions of mass, velocity and radius in it will give the dimension of angular momentum.
$\left[ L \right]=\left[ M \right]\left[ L{{T}^{-1}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-1}} \right]$
The Planck’s constant is having a dimensional formula,
$\begin{aligned} & E=h\nu \\\ & \Rightarrow h=\dfrac{E}{\nu } \\\ \end{aligned}$
Substituting the dimensions of the energy and frequency in it will give,
$\left[ h \right]=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ {{T}^{-1}} \right]}=\left[ M{{L}^{2}}{{T}^{-1}} \right]$
Therefore they are having the same dimensions.
The work can be dimensionally expressed as,
$\left[ W \right]=\left[ F \right]\left[ S \right]$
Substituting the dimensions of the displacement and force in it will give,
$\left[ W \right]=\left[ ML{{T}^{-2}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$
Torque of the body is given by the equation,
$\left[ \tau \right]=\left[ F \right]\left[ r \right]$
Substituting the dimensions of force and radius in it will give,
$\left[ \tau \right]=\left[ ML{{T}^{-2}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$
Therefore they are also having the same dimension.

Hence the correct answer is given as option C.

Note:
The Moment of a force is the amount of its tendency to bring a body into rotation about a specific point or axis. This is not the same as the tendency for a body to move in the direction of the force. Moment of inertia is otherwise known as rotational inertia.