Angular Momentum
Angular Momentum
Rotational Dynamics
Angular momentum is a fundamental quantity in rotational dynamics that describes the extent of rotational motion possessed by a particle or a rigid body about a fixed axis. It is the rotational analog of linear momentum and plays a central role in analyzing systems involving circular or rotational motion.
Definition of Angular Momentum
For a single particle, the angular momentum \(\vec{L}\) with respect to a fixed axis is defined as the vector cross product of its position vector \(\vec{r}\) (relative to the axis) and its linear momentum \(\vec{p}\):
The SI unit of angular momentum is kg·m²/s, and its dimensional formula is \([ML^2T^{-1}]\).
Angular Momentum (): For a particle at position $ \overset{\to }{r} $ with linear momentum $ \overset{\to }{p}$, its angular momentum relative to the origin is defined as the cross product $ \overset{\to }{l}=\overset{\to }{r}\,×\,\overset{\to }{p}$. The right-hand rule (as shown) gives the direction of $ \overset{\to }{l} $, which is perpendicular to both $ \overset{\to }{r} $ and $ \overset{\to }{p} $ (e.g., in the z-direction for motion in the xy-plane).
Angular Momentum of a Point Mass
Consider a point mass \(m\) moving in a circular path of radius \(r\) around a fixed axis. The velocity \(\vec{v}\) is tangential to the circular path, and the angle between \(\vec{r}\) and \(\vec{v}\) is \(90^\circ\). Therefore, the magnitude of angular momentum is:
\[ \text{Since } p = mv, \text{ and } v = r\omega, \text{ we get:} \]
\[ L = r (mv) = m r^2 \omega \]
Thus, for a point mass in circular motion, angular momentum is proportional to the square of the radius and angular velocity.
Angular Momentum of a Rigid Body
Now consider a rigid body composed of several point masses \(m_1, m_2, …, m_n\) located at distances \(r_1, r_2, …, r_n\) from the axis of rotation.
The total angular momentum \(L_{\text{net}}\) of the rigid body is the sum of the angular momenta of all its constituent particles:
where \(I = \sum m_i r_i^2\) is the moment of inertia of the rigid body about the axis.
Illustration of a rigid body with distributed point masses rotating about a central axis.
Relationship Between Torque and Angular Momentum
Angular momentum and torque are related through the time derivative:
This equation is the rotational analog of Newton’s second law. It states that the torque acting on an object equals the time rate of change of its angular momentum.
Comparison of Linear and Rotational Analogs
| Linear Motion | Rotational Motion |
|---|---|
| Force (\( \vec{F} \)) | Torque (\( \vec{\tau} = \vec{r} \times \vec{F} \)) |
| Linear Momentum (\( \vec{p} = m\vec{v} \)) | Angular Momentum (\( \vec{L} = \vec{r} \times \vec{p} \)) |
| Mass (\( m \)) | Moment of Inertia (\( I \)) |
| Linear Acceleration (\( \vec{a} \)) | Angular Acceleration (\( \vec{\alpha} \)) |
| Displacement (\( \vec{x} \)) | Angular Displacement (\( \vec{\theta} \)) |
| Velocity (\( \vec{v} \)) | Angular Velocity (\( \vec{\omega} \)) |
Key Relationships:
- \( \vec{\tau} = I\vec{\alpha} \) (Rotational analog of \( \vec{F} = m\vec{a} \))
- \( \vec{L} = I\vec{\omega} \) (Rotational analog of \( \vec{p} = m\vec{v} \))
- \( \vec{\tau} = \frac{d\vec{L}}{dt} \) (Rotational analog of \( \vec{F} = \frac{d\vec{p}}{dt} \))
Conservation of Angular Momentum
In the absence of an external torque, the total angular momentum of a system remains constant. This is expressed as:
This conservation law explains numerous phenomena in physics and engineering.
Conservation of angular momentum in figure skating: A wider mass distribution (a) results in lower spin speed, which increases sharply (b) as the mass is brought closer to the axis of rotation.
Example: When a skater draws in her arms, her moment of inertia decreases, and thus her angular velocity increases to conserve angular momentum.
A diver conserves angular momentum (L, arrow tail) throughout a parabolic dive, controlling spin rate by changing body configuration.
Application: Gyroscope Behavior
A gyroscope is a rotating wheel mounted to spin about an axis in such a way that it resists orientation changes. The behavior of a gyroscope, such as its ability to remain upright or precess when tilted, is governed by the vector nature of angular momentum.
Gyroscope:
- (a) Gyroscope mounted on an axis
- (b) Gyroscope precession due to torque from gravity
- (c) Vector diagram showing change in direction of angular momentum
When the wheel spins, the system maintains its orientation due to the conservation of angular momentum. If a torque is applied (e.g., due to gravity), the gyroscope’s axis begins to precess — that is, it moves in a circular path.
Activity: Demonstration of Angular Momentum Conservation
🧪 Suggested Setup: Hold a pair of dumbbells while spinning on a rotating platform. Begin with arms close to the body, then extend them outward. Observe the change in spin rate.
Explanation: When arms are extended, the moment of inertia increases and angular velocity decreases to conserve angular momentum (a). Bringing arms in again reverses the effect (b).
Example
Problem:
What is the angular momentum of a \(3.6 \, \text{kg}\) uniform cylindrical grinding wheel of radius \(0.31 \, \text{m}\) rotating at \(1150 \, \text{rpm}\)?
What torque is required to stop it in \(7.8 \, \text{s}\)?
Solution:
(a) Angular velocity:
Moment of inertia of a solid disc:
Angular momentum:
(b) Required torque:
Assignment
Challenge:
Given that Earth rotates about its own axis with angular speed \(7.29 \times 10^{-5} \, \text{rad/s}\), calculate its angular momentum.
Summary of Key Concepts
| Concept | Equation |
|---|---|
| Angular Momentum (point) | \(L = m r^2 \omega\) |
| Angular Momentum (rigid) | \(L = I \omega\) |
| Torque–Angular Momentum | \(\vec{\tau} = \frac{d\vec{L}}{dt}\) |
| Conservation Principle | \(I_1 \omega_1 = I_2 \omega_2\) |
| Precession | \(\Delta \vec{L} = \vec{\tau} \cdot \Delta t\) |
