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 $I<em>1 = 5.33 ext{ kg·m}^2$, then reducing to $I</em>2 = 1.60 ext{ kg·m}^2$:
  • Angular momentum conservation $L<em>1 = L</em>2$ results in angular speed $heta'$ calculation:
  • $I<em>1 heta</em>1 = I<em>2 heta</em>2$ 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:
  • $L = I<em>{initial} heta</em>{initial} = I<em>{final} heta</em>{final}$
  • 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.