JEE Main 2025 — Rotational Motion Question with Solution

Let's examine each expression step by step:

$$\vec{\tau }=\vec{r}\times \vec{L}$$

  Here, $$\vec{L}$$ is the angular momentum. However, torque is defined as the time derivative of angular momentum:

  $$\vec{\tau }=\frac{d\vec{L}}{dt},$$

  not as the cross product of the position vector with the angular momentum. In fact, if you write

  $$\vec{r}\times \vec{L}=\vec{r}\times (\vec{r}\times \vec{p}),$$

  you don't obtain the standard expression for torque. Thus, this expression is not correct.

$$\vec{\tau }=\frac{d}{dt}(\vec{r}\times \vec{p})$$

  For a particle, the angular momentum is defined as

  $$\vec{L}=\vec{r}\times \vec{p}.$$

  Taking the time derivative gives

  $$\frac{d}{dt}(\vec{r}\times \vec{p})=\frac{d\vec{r}}{dt}\times \vec{p}+\vec{r}\times \frac{d\vec{p}}{dt}.$$

  Since $$\frac{d\vec{r}}{dt}=\vec{v}$$ and $$\vec{p}=m\vec{v},$$ the term

  $$\vec{v}\times m\vec{v}$$

  is zero. This simplifies to

  $$\vec{\tau }=\vec{r}\times \frac{d\vec{p}}{dt},$$

  which is a standard expression for torque. So, this expression is correct.

$$\vec{\tau }=\vec{r}\times \frac{d\vec{p}}{dt}$$

  This is the standard definition of torque, as $$\frac{d\vec{p}}{dt}$$ is the net force $$\vec{F}.$$ Hence, we can also write

  $$\vec{\tau }=\vec{r}\times \vec{F}.$$

  This expression is correct.

$$\vec{\tau }=I\vec{\alpha }$$

  This relation applies to rigid bodies rotating about a fixed axis (where the moment of inertia $$I$$ is constant and can be treated as a scalar). It is a common form used in rotational dynamics, although one must be cautious since it is a special case. In the context of this problem, it is acceptable as a correct expression.

$$\vec{\tau }=\vec{r}\times \vec{F}$$

  This is the fundamental definition of torque in physics. It directly relates the force applied to a particle and its lever arm. This expression is clearly correct.

To summarize:

Expression A is not a standard or generally valid expression for torque.

Expressions B, C, D, and E are acceptable under the usual assumptions in mechanics.

Looking at the provided options, the correct answer is:

  Option C: B, C, D and E Only.