Torque and angular momentum
Torque and Angular Momentum
Torque and angular momentum are fundamental concepts in the study of rotational motion in physics. They play a crucial role in understanding how objects rotate and how their rotational motion changes over time. Below, we will explore these concepts in depth, providing a comprehensive understanding suitable for exam preparation.
Torque
Torque, often symbolized by the Greek letter $\tau$, is a measure of the rotational force applied to an object. It is the rotational equivalent of linear force and determines how effectively a force can cause an object to rotate about an axis.
Definition and Formula
Torque is defined as the product of the force applied to an object and the lever arm (the perpendicular distance from the axis of rotation to the line of action of the force):
$$ \tau = r \times F $$
where:
- $\tau$ is the torque
- $r$ is the lever arm (distance from the axis of rotation to the point where the force is applied)
- $F$ is the force applied
Units
The SI unit of torque is the newton-meter (Nm).
Direction
Torque is a vector quantity and has both magnitude and direction. The direction of torque follows the right-hand rule: if you curl the fingers of your right hand in the direction of the force, your thumb points in the direction of the torque.
Angular Momentum
Angular momentum, symbolized by $L$, is the rotational equivalent of linear momentum. It represents the quantity of rotation of a body and is conserved in a closed system where no external torques are applied.
Definition and Formula
Angular momentum is defined as the product of the moment of inertia and the angular velocity:
$$ L = I \omega $$
where:
- $L$ is the angular momentum
- $I$ is the moment of inertia (a measure of an object's resistance to changes in its rotation)
- $\omega$ is the angular velocity
Units
The SI unit of angular momentum is the kilogram meter squared per second (kg·m²/s).
Conservation
The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant.
Differences and Important Points
| Aspect | Torque ($\tau$) | Angular Momentum ($L$) |
|---|---|---|
| Definition | Measure of rotational force | Measure of rotational motion |
| Formula | $\tau = r \times F$ | $L = I \omega$ |
| Units | Newton-meter (Nm) | Kilogram meter squared per second (kg·m²/s) |
| Conservation | Not conserved (changes with applied force) | Conserved in the absence of external torques |
| Dependency | Depends on force and lever arm | Depends on moment of inertia and angular velocity |
| Direction | Follows right-hand rule | Direction of rotation |
Examples
Example 1: Calculating Torque
A wrench is used to tighten a bolt. If the wrench is 0.5 meters long and the mechanic applies a force of 200 N perpendicular to the wrench, what is the torque applied to the bolt?
Using the formula for torque:
$$ \tau = r \times F = 0.5 \, \text{m} \times 200 \, \text{N} = 100 \, \text{Nm} $$
The torque applied to the bolt is 100 Nm.
Example 2: Angular Momentum and Ice Skater
An ice skater is spinning with her arms extended. She has a moment of inertia $I_1$ and an angular velocity $\omega_1$. When she pulls her arms in, she reduces her moment of inertia to $I_2$ but her angular velocity increases to $\omega_2$. If no external torques act on her, her angular momentum is conserved.
Initially: $L_1 = I_1 \omega_1$
Finally: $L_2 = I_2 \omega_2$
By conservation of angular momentum: $L_1 = L_2$
Therefore: $I_1 \omega_1 = I_2 \omega_2$
This explains why the skater spins faster when she pulls her arms in.
Understanding torque and angular momentum is essential for analyzing rotational motion in physics. These concepts are widely applicable, from simple mechanical systems to complex astrophysical phenomena.