Relation between Torque and Angular Momentum

Concept Overview

The relation between torque and angular momentum forms a fundamental connection between rotational dynamics and rotational kinematics, just as force and linear momentum are related in translational motion.

Let a particle of mass [ m ] move in a circle about a fixed point (origin) under the action of a force [ \vec{F} ].
The angular momentum of the particle about the origin is:
[ \vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m \vec{v}) ]

Differentiating both sides with respect to time:

[ \dfrac{d\vec{L}}{dt} = \dfrac{d\vec{r}}{dt} \times m\vec{v} + \vec{r} \times m\frac{d\vec{v}}{dt} ]

But since [ \dfrac{d\vec{r}}{dt} = \vec{v} ] and [ \vec{v} \times \vec{v} = 0 ],
[ \dfrac{d\vec{L}}{dt} = \vec{r} \times m\dfrac{d\vec{v}}{dt} = \vec{r} \times \vec{F} ]

Hence,
[ \boxed{\vec{\tau} = \dfrac{d\vec{L}}{dt}} ]

This means the torque acting on a particle is equal to the time rate of change of its angular momentum.

Physical Interpretation

• If net torque on a system is zero, angular momentum remains conserved.
• Torque acts as the rotational analog of force, while angular momentum corresponds to linear momentum.
• A constant torque causes angular momentum to change linearly with time.

Practice Questions (with Solutions)

Q1. A constant torque of [ 20 , \text{N·m} ] acts on a wheel of moment of inertia [ 4 , \text{kg·m}^2 ]. Find the change in angular momentum in 5 s.
Solution:
[ \tau = \frac{dL}{dt} \Rightarrow dL = \tau , dt ]
[ \Delta L = 20 \times 5 = 100 , \text{kg·m}^2/\text{s} ]

Q2. If torque on a rotating body is zero, what can you say about its angular momentum?
Solution:
From [ \tau = \frac{dL}{dt} ], if [ \tau = 0 ], then [ \frac{dL}{dt} = 0 ].
Hence, [ L = \text{constant} ] — angular momentum is conserved.

Q3. A flywheel increases its angular momentum from [ 5 , \text{kg·m}^2/s ] to [ 25 , \text{kg·m}^2/s ] in 4 s. Find the average torque.
Solution:
[ \tau = \frac{\Delta L}{\Delta t} = \frac{25 – 5}{4} = 5 , \text{N·m} ]

Q4. A rotating disk has moment of inertia [ 0.1 , \text{kg·m}^2 ] and angular velocity increases from [ 20 ] to [ 50 , \text{rad/s} ] in 3 s. Find the torque.
Solution:
[ L_1 = I\omega_1 = 0.1 \times 20 = 2 ]
[ L_2 = I\omega_2 = 0.1 \times 50 = 5 ]
[ \tau = \frac{L_2 – L_1}{t} = \frac{3}{3} = 1 , \text{N·m} ]

Q5. The angular momentum of a system changes by [ 12 , \text{kg·m}^2/s ] in 6 s. Find the torque.
Solution:
[ \tau = \frac{\Delta L}{\Delta t} = \frac{12}{6} = 2 , \text{N·m} ]

Conceptual Questions

FAQ / Common Misconceptions