Which of the following are correct expression for torque acting on a body?
A. τ⃗=r⃗ × L⃗
B. τ⃗=d/(d t)(r⃗ × p⃗)
C. τ⃗=r⃗ × (d p⃗)/(d t)
D. τ⃗=I α⃗
E. τ⃗=r⃗ × F⃗
( r⃗= position vector; p⃗= linear momentum; L⃗= angular momentum; α⃗= angular acceleration; I= moment of inertia; F⃗= force; t= time)
Choose the correct answer from the options given below:

Solution

Physics sir · 3 steps

Torque is r⃗ × F⃗, so test each expression by whether it reduces to that.

1. Start from definition. Torque is τ⃗ = r⃗ × F⃗, and Newton's second law gives F⃗ = dp⃗/dt. Any expression that reduces to r⃗ × F⃗ is correct.

   d/dt(r⃗ × p⃗) = v⃗ × mv⃗ + r⃗ × dp⃗/dt = r⃗ × F⃗

2. Why B works. The first term vanishes because v⃗ and mv⃗ are parallel, and the cross product of parallel vectors is zero. So the rate of change of angular momentum is the torque.

   Expression	Check	Correct?
   A. r⃗ × L⃗	Units are m × kg m² s⁻¹, not N m	No
   B. d/dt(r⃗ × p⃗)	= dL⃗/dt = r⃗ × F⃗	Yes
   C. r⃗ × dp⃗/dt	dp⃗/dt = F⃗	Yes
   D. Iα⃗	Rotational form of F = ma	Yes
   E. r⃗ × F⃗	The definition itself	Yes

3. So. B, C, D and E are correct expressions for torque, and only A is wrong. The answer is (c).

The idea to keep. Torque equals r⃗ × F⃗, which also equals dL⃗/dt and Iα⃗. When a formula looks unfamiliar, check its units against N m first.

Answer from the NTA final answer key. Solution by PrepWiser.