Question

Show that the angular momentum about any point of a single particle moving with constant velocity remains constant throughout the motion.

Short answer

The angular momentum does remain constant for a particle moving with constant velocity because the mass, velocity, and the direction of the cross product between the radius and velocity vectors remain constant.

Step-by-step solution

Express Angular Momentum in Vector Form

Angular momentum, \( L \), can be expressed in vector form as \( L = m(\vec{r} \times \vec{v}) \), where \( m \) is the mass of the particle, \( \vec{r} \) is the position vector of the particle from the point about which we are calculating angular momentum and \( \vec{v} \) is the velocity vector of the particle.

Express Velocity as Constant

Given that the particle is moving with a constant velocity, this implies that \( \vec{v} \) is a constant vector, \( \vec{v} = constant \). Additionally, as \( m \) is a quantity inherent to the particle, \( m \) is also a constant.

Determine the Cross Product

The cross product \( \vec{r} \times \vec{v} \) obtains a vector that is perpendicular to the plane containing \( \vec{r} \) and \( \vec{v} \), following the right-hand rule. As the particle moves, \( \vec{r} \) changes its direction, but the plane defined by \( \vec{r} \) and \( \vec{v} \) remains the same. Therefore, the direction of \( \vec{r} \times \vec{v} \) remains constant even as the particle moves.

Conclude that Angular Momentum Remains Constant

Substituting the constant values of \( m \) and \( \vec{v} \), and the fact that the direction of \( \vec{r} \times \vec{v} \) remains constant into the equation for angular momentum, \( L = m(\vec{r} \times \vec{v}) \), it can be deduced that, \( L \) also remains constant. Hence, the angular momentum of a single particle moving with constant velocity remains constant throughout the motion.

Key concepts

Constant Velocity

When we say a particle is moving with constant velocity, it means both the speed and direction of the particle's motion are not changing over time. Unlike acceleration, where the velocity might change, constant velocity implies a steady-state motion.
This is crucial in physics as it simplifies many calculations, such as those involving forces and momentum. In our context of angular momentum, constant velocity means the velocity vector, \( \vec{v} \), remains unchanged throughout the motion.
Thus, any effect due to changing velocity vectors is effectively not present, allowing certain physical properties, like angular momentum, to be conserved.

Vector Cross Product

The vector cross product is a fundamental operation in physics and mathematics that results in a vector that is perpendicular to the plane formed by the two original vectors.
The cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is represented as \( \vec{a} \times \vec{b} \) and can be calculated as:

• Magnitude: \(|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin(\theta)\)
• Direction: Given by the right-hand rule, ensuring it is perpendicular to both \( \vec{a} \) and \( \vec{b} \)

This operation is especially useful in calculations like those involving torque or angular momentum. In our exercise, the cross product \( \vec{r} \times \vec{v} \) plays a critical role in determining the direction of the angular momentum vector.

Right-Hand Rule

The right-hand rule is a simple way to determine the direction of the resultant vector in a cross product operation.
To use this rule, follow these steps:

• Point your thumb in the direction of the first vector, \( \vec{r} \).
• Extend your fingers in the direction of the second vector, \( \vec{v} \).
• Your palm will naturally face the direction of the resulting vector \( \vec{r} \times \vec{v} \).

This rule helps visualize the orientation of vectors in a three-dimensional space.
For problems involving angular momentum, applying the right-hand rule ensures the direction of vectors like \( \vec{r} \times \vec{v} \) is correctly identified.