6.1 Rotational Kinetic Energy
Just as a moving object possesses translational kinetic energy, a spinning object possesses Rotational Kinetic Energy. Even if the center of mass of the object isn't moving through space (like a spinning ceiling fan), the individual particles making up the object are moving, so there is kinetic energy in the system.
Rotational Kinetic Energy (Krot): The energy an object possesses due to its rotation about an axis.
Rotational Kinetic Energy Formula
- Krot = Rotational Kinetic Energy (Joules, J)
- I = Rotational Inertia (kg·m2)
- ω = Angular velocity (rad/s)
6.2 Torque and Work
We know that in linear dynamics, Work is done when a Force acts over a distance (W = Fd). In rotational dynamics, Rotational Work is done when a Torque acts over an angular displacement.
Rotational Work Formula
- W = Work done by torque (Joules, J)
- τ = Torque (N·m)
- Δθ = Angular displacement (radians)
The Rotational Work-Energy Theorem
Just like the linear Work-Energy Theorem (Wnet = ΔK), the net rotational work done on an object equals its change in rotational kinetic energy.
6.3 Angular Momentum and Angular Impulse
Angular momentum is the rotational analogue of linear momentum (p = mv). It measures how difficult it is to stop a spinning object, or an object moving around a pivot point.
Angular Momentum (L): The rotational momentum of a spinning or orbiting object.
For a Rigid Spinning Object
Used for solid objects spinning about their center (like a top or a wheel).
For a Point Mass
Used when a particle is moving past a specific reference/pivot point.
Angular Impulse
Just as a linear impulse (FΔt) changes linear momentum, an Angular Impulse changes angular momentum. A net external torque applied over a time interval will speed up or slow down an object's rotation.
6.4 Conservation of Angular Momentum
One of the most important principles in rotational dynamics: If the net external torque on a system is zero, the total angular momentum of the system is conserved.
Thought Experiment: The Figure Skater
When a spinning skater pulls their arms inward, they are decreasing their Rotational Inertia (If < Ii). Because there is no external torque (ignoring ice friction), angular momentum (L) must remain constant. To compensate for the lower I, their angular velocity (ω) must increase! This is why they spin faster.
6.5 Rolling
When an object rolls down a hill without slipping, it is doing two things simultaneously: translating through space (moving linearly) AND rotating around its center of mass.
Total Kinetic Energy of Rolling
Because the object is both translating and rotating, its total kinetic energy is the sum of both types.
⚠️ Condition for Rolling Without Slipping:
For true rolling, the translational velocity of the center of mass (v) is perfectly locked to the angular velocity (ω) by the radius (r). The equation v = rω allows you to substitute variables and solve complex energy problems!
6.6 Motion of Orbiting Satellites
We often analyze planets and satellites using rotational dynamics. The gravitational force between a planet and a satellite always acts purely in the radial direction (pointing straight toward the center of mass).
Why is Angular Momentum Conserved in Orbit?
Because the force of gravity is entirely radial, the angle between the force vector and the radius vector is 180° (or 0°).
Torque is τ = rF sin(θ). Since sin(180°) = 0, Gravity produces ZERO torque on an orbiting satellite.
With τnet = 0, the satellite's angular momentum (L = mvr) is perfectly conserved. This means when a satellite is closer to the planet (smaller r), it MUST travel faster (larger v) to keep L constant!
Unit 6 Key Takeaways
Rolling objects share their total energy between translation and rotation (Ktot = Ktrans + Krot).
Use v = rω to substitute out variables in rolling energy problems.
Angular impulse (ΔL = τΔt) is required to change a system's angular momentum.
A point mass moving in a straight line HAS angular momentum relative to an off-axis pivot (L = mvr⊥).
If no net external torque acts, Angular Momentum is conserved (Iiωi = Ifωf).
Gravity exerts zero torque on orbiting objects, conserving their angular momentum.
End of Unit 6 Study Guide.