If a particle with mass m moves with position vector r(t) ,...

If a particle with mass $ m $ moves with position vector $ r(t) $, then its angular momentum is defined as $ L(t) = m r(t) \times v(t) $ and its torque as $ \tau (t) = m r(t) \times a(t) $. Show that $ L^\prime (t) = \tau (t) $. Deduce that if $ \tau (t) = 0 $ for all $ t $, then $ L(t) $ is constant. (This is the law of conservation of angular momentum.)

Key Concepts

Differentiation of the Cross Product

When the angular momentum, defined as a cross product of two time-dependent vectors, is differentiated with respect to time, the product rule for derivatives applies. This leads to a relationship where the time rate of change of angular momentum is directly associated with torque, underscoring the mathematical connection between these two quantities.

Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque acts on a system, the angular momentum remains constant over time. This conservation law is pivotal in many areas of physics, explaining phenomena ranging from the stable rotation of planets to the behavior of spinning ice skaters.

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is defined as the cross product of the position vector and the linear momentum of the object. This concept provides insights into how the distribution of mass relative to a pivot or axis affects the object's ability to rotate and is fundamental in analyzing rotational dynamics.

Torque

Torque is the rotational equivalent of force and is defined as the cross product of the position vector and the force applied (or mass times acceleration in this context). It represents the tendency of a force to cause an object to rotate about an axis or a point, making it a crucial concept in understanding the dynamics of rotational systems.

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