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

To demonstrate that the angular momentum of a single particle moving with constant velocity remains constant about any point, we need to consider the definitions and properties of angular momentum in the context of linear motion. Let's break this down step by step.

Understanding Angular Momentum

The angular momentum (L) of a particle relative to a point (usually referred to as the origin) can be defined as:

L = r × p

where:

• r is the position vector from the point of reference to the particle.
• p is the linear momentum of the particle, given by p = mv, with m being the mass and v the velocity.

Analyzing the Situation

In this scenario, we have a particle moving with a constant velocity. This means that both the speed and direction of the particle's motion do not change over time. Let’s denote the constant velocity as v and the mass of the particle as m.

Position Vector and Linear Momentum

Let’s say at any time t, the position vector of the particle relative to our chosen point is r(t). Since the particle is moving with a constant velocity, we can express its position as:

r(t) = r_0 + vt

where r_0 is the initial position vector at time t = 0.

Calculating Angular Momentum

Now, we can express the angular momentum at time t:

L(t) = r(t) × p = r(t) × mv

Substituting for r(t), we have:

L(t) = (r_0 + vt) × mv

Using the distributive property of the cross product, this expands to:

L(t) = (r_0 × mv) + (vt × mv)

Notice that (r_0 × mv) is a constant vector since both r_0 and m are constants, and the velocity v does not vary. The term (vt × mv) also evaluates to zero because the cross product of a vector with itself is always zero.

Conclusion on Angular Momentum's Constancy

Therefore, we can simplify the angular momentum expression to:

L(t) = constant

This shows that the angular momentum about any point remains constant throughout the motion of the particle. The constancy arises from the fact that the linear momentum is constant (due to constant velocity), and the position vector's changes do not alter the angular momentum since they do not affect the cross product in a way that changes its magnitude or direction.

In summary, when analyzing a particle in uniform motion, the angular momentum remains conserved about any point, highlighting a fundamental principle in physics regarding rotational dynamics and the conservation of angular momentum.