Recall the standard expressions for torque

The torque on a particle about an origin is $\vec{\tau}=\vec{r}\times \vec{F}$ Since $\vec{F}=\frac{d\vec{p}}{dt},$ we also have $\vec{\tau}=\vec{r}\times \frac{d\vec{p}}{dt}.$

Relation with angular momentum

Angular momentum of a particle is $\vec{L}=\vec{r}\times \vec{p}.$ Differentiating,

Check each statement

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

This is incorrect. Torque is not given by $\vec{r}\times \vec{L}$. Also, dimensionally:

• $[\tau]=ML^2T^{-2}$
• $[\vec{r}\times \vec{L}]=L\cdot ML^2T^{-1}=ML^3T^{-1}$

So A is false.

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

Since $\vec{L}=\vec{r}\times \vec{p}$ and $\vec{\tau}=\dfrac{d\vec{L}}{dt}$, this is correct.

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

Since $\dfrac{d\vec{p}}{dt}=\vec{F}$, this is correct.

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

This is valid for rotation about a fixed axis when $I$ is constant. In standard JEE context, this is taken as a correct expression for torque. So D is correct.

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

This is the basic definition of torque, so E is correct.

Correct set

Correct expressions are: $B,\ C,\ D,\ E$

Match with options

This corresponds to Option C.