rotational motion

Rotational Motion Study Notes

Learning Competencies

  • Describe rotational motion in terms of the following aspects:

    • Angular displacement

    • Angular velocity

    • Angular acceleration

    • Angular frequency

Inertia

  • Definition: Inertia is the property of matter to resist changes in its state of motion.

    • An object at rest tends to stay at rest.

    • An object in motion tends to continue moving in a straight line at a constant velocity.

  • Similar principles apply to objects in rotational motion.

Moment of Inertia

  • Definition: Moment of inertia, also known as rotational inertia, is defined as the property of a rotating body to resist change in its state of rotation.

  • SI Unit: The unit of moment of inertia is ext{kg} imes ext{m}^2.

Behavior of Rotating Objects

  • An object rotating about an axis will continue rotating about that axis unless acted upon by an unbalanced external force (torque).

  • This resistance to change in motion is physically embodied in the mass of the object.

Dependence of Moment of Inertia

  • The moment of inertia depends on the distribution of mass:

    • A mass farther from the axis of rotation has a greater moment of inertia than the same mass that is closer to the rotation axis.

  • Example:

    • Dumbbell A vs. Dumbbell B

    • Dumbbell A has more distributed mass away from its axis, making it harder to rotate than Dumbbell B, which has its masses closer to the axis of rotation.

Measurement of Moment of Inertia

  • The moment of inertia provides a measurement of the body's resistance to changes in its rotational motion:

    • A larger moment of inertia means it is harder to initiate or stop rotational motion.

Moment of Inertia of a Single Particle

  • The moment of inertia (I) of a particle around an axis can be calculated using the formula: I = m imes r^2

    • Where:

    • I = moment of inertia

    • m = mass of the particle

    • r = distance from the axis of rotation

Moment of Inertia for Multiple Particles

  • For a system of multiple particles, the total moment of inertia is the sum of individual moments of inertia:
    I = m1r1^2 + m2r2^2 + m3r3^2 + ext{…}

Radius of Gyration

  • Definition: The radius of gyration (k) is the distance from an axis of rotation where the body's mass is assumed to be concentrated without changing its moment of inertia about that axis.

  • Formula:
    k = \frac{I}{m}

  • Analogy: It is analogous to the center of mass.

Importance of Axis of Rotation

  • It is crucial to specify the axis of rotation when discussing moment of inertia because it changes depending on the axis. Each axis results in a different moment of inertia.

Application of Calculus

  • Note: Calculus is often used for complex calculations of moment of inertia; however, simple formulas exist for symmetrical bodies.

Moments of Inertia for Common Shapes

  • Rod about Center:
    I = \frac{1}{12} ml^2

  • Rod or Disc about Axis:
    I = \frac{1}{2} mr^2

  • Sphere:
    I = \frac{2}{5} mr^2

  • Hollow Rod or Disc (Thin Wall) about Axis:
    I = mr^2

Example Problem: Moment of Inertia of a Solid Cylinder

  • Given:

    • Mass (m) = 4 kg

    • Diameter = 1.0 m

    • Radius (r) = \frac{1.0}{2} = 0.50 m

  • Shape: Solid cylinder (refer to Figure A)

  • Axis of rotation: Through its center

  • Find: Moment of Inertia (I)

Example Solution

  • Formula:
    I = \frac{1}{2} mr^2

  • Calculations:
    I = \frac{1}{2} (4 ext{ kg}) (0.50 ext{ m})^2
    I = 0.50 ext{ kg-m}^2