Torque, Angular Momentum, and Moment of Inertia Review

Torque, Angular Momentum, and Moment of Inertia Review

Key Concepts

  • Torque: The rotational analog of linear force, calculated as au = r imes F where au is the torque, r is the radius vector, and F is the force.

  • Angular Momentum (L): The product of the moment of inertia (I) and the angular velocity ( heta) of a rotating object, expressed as L = I heta.

  • Moment of Inertia (I): A measure of an object's resistance to changes in its rotational motion, defined as I = ext{∑} m r^2 where m is the mass and r is the distance from the axis of rotation.

Moment of Inertia Derivations

1. Rod Spinning About Its Midpoint

  • Formula: I = rac{1}{12} ml^2 Where:
    • m = mass of the rod
    • l = length of the rod

Derivation:

  • Let linear mass density be rac{m}{L} and the differential mass dm = rac{m}{L} dx.
  • Moment of inertia I = ext{∫} r^2 dm:
    • I = ext{∫}_{-l/2}^{l/2} x^2 rac{m}{L} dx
    • Simplifying leads to I = rac{1}{12} ml^2.

2. Rod Spinning About Its End

  • Formula: I = rac{1}{3} ml^2

Derivation:

  • Similar to the above: the limits change to [0, l] for the integration:
    • I = ext{∫}_0^l x^2 rac{m}{L} dx
    • Compute the integral to arrive at I = rac{1}{3} ml^2.

3. Uniform Disk Spinning About Its Center

  • Formula: I = rac{1}{2} mr^2

Derivation:

  • Use area mass density rac{M}{A} where A = ext{π}r^2:
    • dm = ext{σ} dA = ext{σ}(2 ext{π}r dr)
    • Resulting in integral I = ext{∫} r^2 dm up to R yields I = rac{1}{2} mr^2.

Examples

Example 1: Ceiling Fan Moment of Inertia

  • Angular speed is 2.75 ext{ rad/s}; frictional torque au = 0.120 ext{ N·m}; time t = 22.5 ext{ s}. Compute the moment of inertia using:
    • Variables:
    • Angular acceleration rac{ ext{d} heta}{ ext{d}t} = rac{2.75}{22.5}
    • I = rac{ au}{ ext{acceleration}} gives I.

Example 2: Angular Speed Change on Spinning Stool

  • Initially I1 = 5.33 ext{ kg·m}^2, then reducing to I2 = 1.60 ext{ kg·m}^2:
    • Angular momentum conservation L1 = L2 results in angular speed heta' calculation:
    • I1 heta1 = I2 heta2 implies heta2 = rac{I1 heta1}{I2}.

Example 3: Merry-Go-Round with Person Jumping On

  • The initial and the final angular speeds relate through conservation of angular momentum:
    • L = I{initial} heta{initial} = I{final} heta{final}
    • Calculate combined system moment of inertia as I + m r^2 after the person jumps on.

Additional Calculations and Concepts

  • Parallel Axis Theorem: If you know the moment of inertia about an axis through the center of mass, you can find it about any axis parallel to it:
    • I' = I_{cm} + md^2
  • Calculation for a Rod on a Spinning Framework: Provides practice with inertia calculations on different axes.

Conclusion

  • Mastery of the moments of inertia and associated principles is crucial for understanding rotational dynamics. Continuous practice with various problem setups reinforces these concepts.