# Identify which of the following option is possible when the torque acting on a rigid body under the application of a force is zero

A. Linear momentum is conserved

B. Angular momentum is not conserved

C. Energy is conserved

D. Angular momentum is conserved

Answer

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**Hint:**When force acting on the body is zero, its linear momentum is conserved as force is rate of change of linear momentum. Similarly torque is the rate of change of angular momentum and can be said as the rotational analogue of force.

**Complete step by step answer:**

Let us first talk about force and linear momentum. We know from Newton's 2nd law of motion that the force $F$ acting on the body is the time rate of change of linear momentum $p$ of that body. This can be mathematically represented as $F = \dfrac{{dp}}{{dt}}$ . From this expression we can say that when force acting on a body is zero then its linear momentum is conserved.

Now, we will discuss the relation between torque and angular momentum.

As we know that torque is a rotational analogue of force and has a similar relation with angular momentum as that between force and linear momentum.

Mathematically, angular momentum $\vec L$ can be written as $\vec L = \vec r \times \vec p$ where $\vec r$ is the position vector of the body from any reference and $\vec p$ is the linear momentum of the body.

Now, differentiating both sides of the equation with respect to $t$ , we have

$\dfrac{{d\vec L}}{{dt}} = \dfrac{{d\vec r}}{{dt}} \times \vec p + \vec r \times \dfrac{{d\vec p}}{{dt}}$

Now, we know that $\dfrac{{d\vec r}}{{dt}} = \vec v$ the velocity of the body and as the direction of linear momentum is in the direction velocity, so the first term of the equation becomes zero and we get

$\dfrac{{d\vec L}}{{dt}} = \vec r \times \dfrac{{d\vec p}}{{dt}} = \vec r \times \vec F$

Now, we know that torque is given by $\vec \tau = \vec r \times \vec F$ so finally we have

$\dfrac{{d\vec L}}{{dt}} = \vec \tau $

Therefore we can say from the above equation that when the torque acting on a body is zero, angular momentum of that body is conserved.

**Hence, option D is correct.**

**Note:**Angular momentum has plenty of applications seen in real life such as the movement of an ice-skater when he or she spins produces angular momentum and Gyroscope which is used in space applications also works on the principle of angular momentum.

Last updated date: 30th May 2023

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