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:

1. θ = θ₀ + ωt + (1/2)αt²

   • θ₀ = initial angular position

2. 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