Which of the following are correct expression for torque acting on a body? A. $\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{L}}$ B. $\vec{\tau}=\frac{d}{d t}(\vec{r} \times \vec{p})$ C. $\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \overrightarrow{\mathrm{p}}}{\mathrm{dt}}$ D. $\vec{\tau}=I \vec{\alpha}$ E. $\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}$ ( $\overrightarrow{\mathrm{r}}=$ position vector; $\overrightarrow{\mathrm{p}}=$ linear momentum; $\overrightarrow{\mathrm{L}}=$ angular momentum; $\vec{\alpha}=$ angular acceleration; $\mathrm{I}=$ moment of inertia; $\overrightarrow{\mathrm{F}}=$ force; $\mathrm{t}=$ time) Choose the correct answer from the options given below :

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Rotational Mechanics

Which of the following are correct expression for torque acting on a body?
A.

τ = r \times L

τ = \frac{d}{d t} (r \times p)

τ = r \times \frac{d p}{dt}

τ = I α

τ = r \times F

(

r =

position vector;

p =

linear momentum;

L =

angular momentum;

α =

angular acceleration;

I =

moment of inertia;

F =

force;

t =

time)
Choose the correct answer from the options given below :

Answer: (C)

Angular momentum is

L = r \times p

, and torque is its time rate of change,

τ = \frac{d L}{d t} = \frac{d}{d t} (r \times p)

. Expanding,

\frac{d}{d t} (r \times p) = v \times p + r \times \frac{d p}{d t} = r \times F

, since

v \times p = 0

. For rotation about a fixed axis this equals

I α

as well. The valid expressions are therefore

\frac{d}{d t} (r \times p)

r \times \frac{d p}{d t}

I α

and

r \times F

, while

r \times L

is not a torque.

JEE Main · 2025 · 04 Apr 2025 · Shift 1 · Single correct option