JEE Physics 2025 Question with Solution

Given: We must identify which listed expressions are valid forms of torque acting on a body.

Find: The correct option containing all valid expressions for torque.

Torque can be written in equivalent forms as

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

Also, using rotational dynamics for a rigid body about a fixed axis,

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

Since

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

we get

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

And because

$$\vec{F} = \frac{d\vec{p}}{dt}$$

we also have

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

Now check each statement:

• A is incorrect because torque is not $\vec{r} \times \vec{L}$.
• B is correct because $\vec{L} = \vec{r} \times \vec{p}$.
• C is correct because $\frac{d\vec{p}}{dt} = \vec{F}$.
• D is correct for rigid body rotation about a fixed axis.
• E is correct because torque is $\vec{r} \times \vec{F}$.

Therefore, the correct expressions are B, C, D and E only.

The correct option is C.

Given: The listed expressions relate torque with position vector, momentum, angular momentum, force, and angular acceleration.

Find: Which of the statements A to E are valid.

Step 1: Test statement A.

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

Torque is the time derivative of angular momentum, not the cross product of position vector with angular momentum. So A is incorrect.

Step 2: Test statement B. Using

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

Differentiate with respect to time:

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

So B is correct.

Step 3: Test statement C. Since

$$\frac{d\vec{p}}{dt} = \vec{F}$$

we get

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

So C is correct.

Step 4: Test statement D. For rotational motion of a rigid body about a fixed axis,

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

Hence D is correct.

Step 5: Test statement E. The fundamental definition of torque is

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

Hence E is correct.

Thus the correct set is B, C, D and E only, which matches option C.