Question

The dimensional formula for impulse is same as the dimensional formula for
A. Momentum
B. Force
C. Rate of change of momentum
D. Torque

Answer 1

Recall the dimensional formula for impulse. Recall the formula for each of the physical quantities given in the options and determine the unit of each of these physical quantities. From the units of these physical quantities, determine the dimensional formula for all these quantities. Compare these dimensional formulae with the dimensional formula for impulse.

Complete step by step solution:
The dimensional formula for impulse is $\left[ {ML{T^{ - 1}}} \right]$.
The force acting on an object is given by
$F = ma$
From the above equation, the unit of force becomes kilogram meter per second square.
Therefore, the dimensional formula for force is $\left[ {ML{T^{ - 2}}} \right]$ which is not same as dimensional formula for impulse. Hence, the option B is incorrect.

The rate of change of momentum is given by $\dfrac{{\Delta P}}{{\Delta t}}$.
The unit of rate of change of momentum is kilogram meter per second square. Therefore, the dimensional formula for rate of change of momentum is $\left[ {ML{T^{ - 2}}} \right]$ which is not same as dimensional formula for impulse. Hence, the option C is incorrect.

The torque is given by the product of force and perpendicular distance between the point at which the torque acts and the point of action of force.
$\tau = Fr$
From the above equation, the unit of torque is kilogram meter square per second square.
Therefore, the dimensional formula for torque is $\left[ {M{L^2}{T^{ - 2}}} \right]$ which is not same as dimensional formula for impulse. Hence, the option D is incorrect.

The momentum of an object is given by
$P = mv$
From the above equation, the dimensional formula for momentum is $\left[ {ML{T^{ - 1}}} \right]$ which is the same as the dimensional formula for impulse.

Hence, the correct option is A.

Note: One can also determine the physical quantity having the same dimensional formula as that of the impulse. The impulse acting on an object is equal to the change in linear momentum of that object. The dimensions of linear momentum and change in linear momentum are the same. Hence, the dimensions of impulse will be the same as that of the momentum.