Unit 6: Energy and Momentum of Rotating Systems

Rotational Kinetic Energy

Energy of an object in motion around an axis, similar to linear kinetic energy.

Angular Displacement (θ)

The angle through which an object has rotated about an axis.

Angular Velocity (ω)

The rate of change of angular displacement per unit of time.

Angular Acceleration (α)

The rate of change of angular velocity per unit of time.

Torque (τ)

The rotational equivalent of force, causing an object to rotate.

Moment of Inertia (I)

A measure of an object's resistance to angular acceleration, dependent on mass distribution.

Torque Formula

τ = rF sinθ, where r is the distance from the axis of rotation, F is the applied force, and θ is the angle.

Newton’s 2nd Law for Rotation

τₙₑₜ = Iα, relating net torque to angular acceleration.

Angular Momentum (L)

The product of moment of inertia and angular velocity, L = Iω.

Angular Impulse

The change in angular momentum resulting from torque applied over a period of time, ΔL = τΔt.

Conservation of Angular Momentum

Angular momentum remains constant if no external torque acts: Li = Lf.

Rolling Motion

When an object rotates about an axis while translating across a surface without slipping.

Condition for Rolling without Slipping

Velocity at the bottom point of a rolling object is zero (V = rω).

Total Mechanical Energy in Isolated Systems

The sum of translational, rotational, and gravitational potential energy, E = 1/2mv² + 1/2Iω² + mgh.

Angular Momentum of Satellites

Angular momentum (L = mvr) is conserved for satellites in orbit.

Orbital Velocity of a Satellite

The velocity needed for a satellite to maintain orbit, primarily determined by gravitational force.

Rolling Kinetic Energy Equation

K = 1/2mv² + 1/2Iω², representing total kinetic energy during rolling.

Power in Rotation

P = τω, which indicates how quickly work is done in rotating systems.

Gravitational Force in Orbits

Acts as centripetal force for satellites moving in circular or elliptical orbits.

Example of Conservation of Angular Momentum

Figure skater pulls arms in to spin faster, demonstrating I↓ leads to ω↑.