Define angular momentum (L⃗) and torque (τ⃗). Show that dL⃗/dt = τ⃗.

**OR**

A uniform solid sphere of mass M and radius R rolls down an inclined plane making an angle θ with the horizontal without slipping. Show that the acceleration of the sphere is (5/7)g sin θ. Given: moment

Question

Define angular momentum (L⃗) and torque (τ⃗). Show that dL⃗/dt = τ⃗.

**OR**

A uniform solid sphere of mass M and radius R rolls down an inclined plane making an angle θ with the horizontal without slipping. Show that the acceleration of the sphere is (5/7)g sin θ. Given: moment of inertia of the sphere is (2/5)MR².

Accepted Answer

>

> [!TLDR]
> Differentiate L = r × p using the product rule; the velocity-cross-momentum term vanishes, leaving exactly the torque.
>
> **Angular momentum:** $\vec L=\vec r	imes\vec p$, the moment of linear momentum about a point.
>
> **Torque:** $\vec	au=\vec r	imes\vec F$, the moment of force about the same point.
>
> **Derivation of $\dfrac{d\vec L}{dt}=\vec	au$:**
> $$\frac{d\vec L}{dt}=\frac{d}{dt}(\vec r	imes\vec p)=\frac{d\vec r}{dt}	imes\vec p+\vec r	imes\frac{d\vec p}{dt}$$
> $$=\vec v	imes m\vec v+\vec r	imes\vec F$$
> …

Concept understanding — Angular Momentum

Angular Momentum: The Rotational Cousin of Momentum

The Two Faces of Angular Momentum

Why Angular Momentum Matters

Connecting the Two Definitions

A Quick Check on Direction …

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