Establish the relation between angular momentum and torque.

**OR**

State the principle of conservation of angular momentum. Apply this principle to explain the following observations: i) An ice-skater can increase her angular velocity by folding her arms and bringing the stretc

Question

Establish the relation between angular momentum and torque.

**OR**

State the principle of conservation of angular momentum. Apply this principle to explain the following observations: i) An ice-skater can increase her angular velocity by folding her arms and bringing the stretched legs closer to the other leg. ii) The angular speed of planets increases, when it comes closer to the sun.

Accepted Answer

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> [!TLDR]
> Differentiating the definition of angular momentum $\vec L = \vec r	imes\vec p$ with respect to time yields $\vec	au = d\vec L/dt$.
>
> For a particle of momentum $\vec p$ at position $\vec r$ (measured from a chosen origin), the angular momentum about that origin is defined as $\vec L = \vec r	imes\vec p$.
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> Differentiating with respect to time using the product rule:
>
> $$\dfrac{d\vec L}{dt} = \dfrac{d\vec r}{dt}	imes\vec p + \vec r	imes\dfrac{d\vec p}{dt}$$
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> The first term is $\vec v	imes\vec p = \vec v	imes(m\vec v) = m(\vec v	imes\vec v) = 0$, since a vector crossed with itself is zero.
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> The second term uses Newton's second law, $\dfrac{d\vec p}{dt}=\vec F$ (the net force on the particle), so it becomes $\vec r	imes\vec F$, which is by definition the torque $\vec	au$ about the same origin.
> …

Concept understanding — Torque

What is Torque? The Intuition

The Precise Definition

Why Torque Equals Rate of Change of Angular Momentum

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