Angular Momentum and Conservation

This lesson covers angular momentum, its conservation, and applications to collision and rotation problems.

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1 Definition of Angular Momentum

The angular momentum $L$L of a particle about an axis is:

$$L = mvr$$L=mvr

where $r$r is the perpendicular distance from the axis to the line of motion of the particle.

For a rigid body rotating about a fixed axis:

$$\boxed{L = I\omega}$$L=Iω​

Property	Detail
SI unit	kg m$^2$2 s$^{-1}$−1 (or N m s)
Vector	Direction along the axis of rotation (right-hand rule)
Depends on	Moment of inertia and angular velocity

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2 Relationship to Torque

Angular momentum is related to torque by:

$$\boxed{\tau = \frac{\mathrm{d}L}{\mathrm{d}t}}$$τ=dtdL​​

This is the rotational analogue of $F = \mathrm{d}p/\mathrm{d}t$F=dp/dt.

If $I$I is constant: $\tau = I\alpha$τ=Iα.

If $\tau = 0$τ=0: $L$L is constant — this gives us conservation of angular momentum.

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3 Conservation of Angular Momentum

When no external torque acts on a system, the total angular momentum is conserved:

$$\boxed{I_1 \omega_1 = I_2 \omega_2}$$I1​ω1​=I2​ω2​​