Animated Solution for Physics - Rotational Motion: Assertion If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass remains constant. Reason The linear momentum of an isolated system remains constant.

Imagine you are standing in the vast emptiness of space, floating next to a massive, irregularly shaped asteroid. You reach out and push it exactly at its center of mass. What happens? Does it spin? Does it move? This simple thought experiment is the key to unlocking one of the most common traps in rotational mechanics.

Many students intuitively believe that if a body isn't spinning, it must be in a state of complete equilibrium. But as we will see, rotational equilibrium and translational equilibrium are two very different beasts. Let's break down the physics behind this fascinating problem.

Torque vs. Force

The assertion claims that if there is no external torque on a body about its center of mass, its velocity must remain constant. To test this, we need to look at the mathematical definition of torque. Torque is the "turning effect" of a force, defined by the cross product:

Here, $\vec{r}$ is the position vector from the axis of rotation to the point where the force is applied. Now, what if we apply a force $\vec{F}$ exactly at the center of mass? In this specific case, the distance $\vec{r}$ is zero. Consequently, the torque $\vec{\tau }$ is also zero. The body will not experience any angular acceleration; it will not spin.

However, this is where the trap snaps shut! Just because the torque is zero does not mean the applied force is zero. We are still exerting a net external force on the body. According to Newton's Second Law of Motion for a system of particles, the net external force governs the translational motion of the center of mass:

Since ${\vec{F}}_{ext}$ is non-zero, the acceleration of the center of mass, ${\vec{a}}_{cm}$, must also be non-zero. If a body is accelerating, its velocity is changing. Therefore, the velocity of the center of mass cannot remain constant. The assertion is fundamentally flawed and is False.

The Sanctity of an Isolated System

Now, let's turn our attention to the reason statement, which introduces the concept of an "isolated system." In physics, an isolated system is a collection of matter that does not interact with the rest of the universe. By definition, the net external force acting on an isolated system is strictly zero.

Let's apply Newton's Second Law in its most general form, relating force to momentum:

For an isolated system, we know that ${\vec{F}}_{ext}=0$. Substituting this into our equation gives:

If the derivative of a quantity with respect to time is zero, that quantity must be a constant. Therefore, the total linear momentum $\vec{p}$ of the isolated system remains perfectly constant. This is the universally revered Law of Conservation of Linear Momentum. The reason statement is a factual, undeniable truth of nature.

We have systematically dismantled the problem. The assertion falls apart because it confuses zero torque with zero force, failing to account for pure translational acceleration. Meanwhile, the reason stands strong as a fundamental conservation law.

Thus, the assertion is false, and the reason is true. The correct choice is Option (d). Always remember: a push at the center of mass won't make a body spin, but it will certainly make it move!