Unit 6: Energy and Momentum of Rotating Systems

You’ll explore the energy and momentum of an object rotating around an axis and connect those concepts to their linear analogs.

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6.1 Rotational Kinetic Energy

Rotational motion is described using:

Angular Displacement (θ)
Angular Velocity (ω)
Angular Acceleration (α)

Key Equations:

θ = θ₀ + ωt + (1/2)αt²
- θ₀ = initial angular position
ω = ω₀ + αt
- ω₀ = initial angular velocity

These are analogous to linear motion equations:

x = x₀ + vt + (1/2)at²
v = v₀ + at

Rolling Motion:

Objects that roll without slipping combine rotation and translation.
The bottom point of a rolling object is momentarily at rest relative to the surface.
Torque (τ) is the rotational equivalent of force.

6.2 Torque and Work
Torque Formula:

τ = rF sinθ

r = distance from axis of rotation
F = force applied
θ = angle between r and F

Newton’s 2nd Law for Rotation:

τₙₑₜ = Iα

I = moment of inertia (resistance to rotational acceleration)

Moment of inertia is a measure of how difficult it is to rotate an object:

For a point mass: I = mr²
For extended objects: depends on shape and axis; use known formulas (e.g., for rods, disks)
Rotational work:
W = τθ
Power in rotation:
P = τω
- Tells how quickly work is done in rotating systems

6.3 Angular Momentum and Angular Impulse

Angular momentum (L):

L = Iω
Depends on mass distribution and angular velocity

Angular impulse is the product of torque and time:

ΔL = τΔt

It equals the change in angular momentum

Newton’s 2nd Law for Rotation:
τ_net=dL/dt

Real-world example: A spinning wheel speeds up or slows down depending on the net torque applied.

6.4 Conservation of Angular Momentum

Angular momentum is conserved when no external torque acts:
L_i = L_f OR I_iω_i = I_fω_f
Figure skater example:
- Pulling arms in → I↓, so ω↑
- Extending arms → I↑, so ω↓
Total mechanical energy is conserved in isolated systems:

E = 1/2mv² + 1/2Iω² + mgh

Includes translational, rotational, and gravitational potential energy

6.5 Rolling

Occurs when an object rotates while translating along a surface without slipping
Key condition for rolling without slipping:
V = rω
At the bottom of the object, velocity = 0 due to the opposite motion from translation and rotation
Total kinetic energy for rolling:

K = 1/2mv² + 1/2Iω²

6.6 Motion of Orbiting Satellites

Satellites move in circular or elliptical orbits due to gravitational force acting as centripetal force:

Orbital velocity of a satellite:

Total mechanical energy in orbit:

Negative energy → bound orbit

Angular momentum of satellite is conserved (assuming no external torques):

L = mvr

Real-world applications: Satellites, planets, moons orbiting due to gravitational forces