Question

How are torque and angular momentum related?

Answer 1

Torque and angular momentum are closely related concepts in rotational dynamics.

1. **Definitions**:
- **Torque (τ)**: Torque is a measure of the rotational force applied around an axis. It is defined mathematically as the cross product of the lever arm vector (\( \vec{r} \)) and the force vector (\( \vec{F} \)):
\[
\vec{\tau} = \vec{r} \times \vec{F}
\]
- **Angular Momentum (L)**: Angular momentum is a measure of the quantity of rotation of an object and is defined as the product of the moment of inertia (I) and the angular velocity (ω):
\[
\vec{L} = I \vec{\omega}
\]
Alternatively, for a point mass, angular momentum can be expressed as:
\[
\vec{L} = \vec{r} \times \vec{p}
\]
where \( \vec{p} = m \vec{v} \) is the linear momentum.

2. **Relationship**:
The relationship between torque and angular momentum can be expressed through the following equation:
\[
\vec{\tau} = \frac{d\vec{L}}{dt}
\]
This equation states that the torque acting on an object is equal to the rate of change of its angular momentum. In other words, if a net torque is applied to an object, it will cause a change in the angular momentum of that object over time.

3. **Implications**:
- If the net torque on a system is zero, then the angular momentum remains constant (conservation of angular momentum).
- Conversely, if a net torque is applied (non-zero), the angular momentum will change according to the amount and direction of the torque.

This relationship shows that torque is responsible for changing the angular motion of an object, much like how force affects linear motion.