Q.State the principle of conservation of angular momentum. Use it to explain why an ice skater who is spinning with arms outstretched spins noticeably faster the moment the arms are pulled in close to the body.

Angular momentum and mechanical energy are conserved; linear momentum is not. Answer (A).

When a body moves under the gravitational attraction of another (central) body:

• The force is central (directed along the line joining them), so torque about the centre is zero -> angular momentum is conserved.
• Gravity is conservative, so total mechanical energy is conserved. …

Motion under a central force stays in a plane because angular momentum is conserved. Answer (B).

A central force always points toward (or away from) a fixed centre, so its torque about that centre is zero (r and F are parallel/anti-parallel, r x F = 0).

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Kepler's second law (the radius vector sweeps equal areas in equal times) is a direct geometric consequence of angular momentum being conserved in a planet's orbit around the Sun.

Gravitational force on a planet from the Sun always acts along the line joining them (a central force), so it produces zero torque about the Sun. With zero net torque, the planet's angular momentum L = m v r sin(theta) about the Sun stays constant throughout the orbit.

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Angular momentum in uniform circular motion is conserved only about the centre of the circle, because the centripetal force passes through that point and produces zero torque there.

For a particle moving in a circle of radius r with constant speed, the only force acting is the centripetal force F, which is always directed radially inward, straight at the centre of the circle.

Torque about any point O is tau = r' x F, where r' is the position vector from O to the particle. If O is the centre of the circle, r' is exactly along the same (or opposite) line as F, so r' x F = 0 at every instant -- zero torque, hence angular momentum about the centre stays constant (both in magnitude, m v r, and direction, perpendicular to the plane of motion).

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The gravitational force of the Sun on a planet always acts along the line joining the Sun and the planet (a central force), so it produces zero torque about the Sun.

Since $\vec\tau = \dfrac{d\vec L}{dt}=\vec r\times \vec F$, and $\vec F$ is always anti-parallel to $\vec r$ (radial), $\vec\tau=0$, so the angular momentum $\vec L$ of the planet about the Sun st …

Planetary motion is a direct illustration of conservation of angular momentum, since gravity is a central force.

The Sun's gravitational pull on a planet always points from the planet towards the Sun, i.e. along the position vector $\vec r$ of the planet (measured from the Sun). Torque is $\vec\tau = \vec r \times \vec F$; since $\vec F$ is anti-parallel to $\vec r$, this cross product is zero, so the torque on the planet about the Sun is always zero. Because $\vec\tau = d\vec L/dt = 0$, the angular momentum $\vec L$ of the planet about …

Halving Earth's radius shortens the day to 6 hours.

Earth spins with angular momentum L = Iω. With no external torque and unchanged mass, L is conserved.

Moment of inertia of a solid sphere I = (2/5)MR^2 ∝ R^2. If R → R/2, then I → I/4.

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