Classical Dynamics Notes

Rigid Body Motion

Characterized by center of mass motion and rotation about the center of mass, with angular velocity ω.
- How to analyze:
  1. Separate the Motion: Break down the rigid body's motion into two components:
    - Translational Motion: This is the movement of the center of mass. Treat the entire body as if it were a point mass located at the center of mass, and analyze its motion using Newton's laws.
      - For example, if the rigid body is moving under the influence of external forces, calculate the net force acting on the body. Then, use Newton's Second Law ( $F = ma$ ) to determine the acceleration of the center of mass. From there, you can find the velocity and position of the center of mass as functions of time.
    - Rotational Motion: This is the rotation of the body about its center of mass. Analyze this motion by considering torque, angular momentum, and the moment of inertia.
      - For instance, If the rigid body is rotating due to an applied torque, calculate the net torque acting on the body. Then, use the rotational analogue of Newton's Second Law ( $N = Iα$ ) to find the angular acceleration. Next, using kinematic equations, determine how angular velocity changes over time.
  2. Define Angular Velocity (ω): Recognize that all points on the rigid body rotate with the same angular velocity about an axis.
    - The direction indicates the axis of rotation, and the magnitude gives the rate of rotation.
  3. Calculate Moments of Inertia: Determine the moment of inertia (I) about the relevant axis of rotation. The moment of inertia depends on the shape and mass distribution of the rigid body.
    - Use standard formulas for common shapes (e.g., sphere, rod, disk) or the parallel axis theorem if the rotation axis is not through the center of mass.
  4. Apply Equations of Motion: Use equations relating torque, angular momentum, and angular velocity to analyze the rotational motion.
    - Newton’s Second Law for rotation: $N = Iα$ where $N$ is the net torque, $I$ is the moment of inertia, and $α$ is the angular acceleration. If torque is constant, angular acceleration is also constant, simplifying the analysis.
    - The relationship between angular momentum & torque: $N = \frac{dL}{dt}$ , where $L$ is the angular momentum. If there's no external torque, angular momentum is conserved, meaning $L$ remains constant.
    - Kinetic Energy (T) = $\frac{1}{2} I ω^2$ , which implies if the moment of inertia decreases, angular velocity increases to conserve angular momentum.
  5. Solve for Unknowns: Combine translational and rotational equations of motion to solve for unknown quantities, such as forces, accelerations, velocities, or angular velocities.
Torque and Angular Momentum: Total angular momentum is conserved for isolated systems with central forces. $L = \sumi ri × p_i$ and $N = \dot{L}$
- Example: A spinning figure skater. When the skater pulls their arms inward, their moment of inertia decreases, and their angular velocity increases to conserve angular momentum.
Parallel Axis Theorem: $\sum I1 → I1 + m\ell^2$
- Application: Calculating the moment of inertia of an object about an axis parallel to an axis through its center of mass. If you know the moment of inertia about the center of mass ( $I_1$ ), you can find it about the parallel axis.
Kinetic Energy: $T = \frac{1}{2} I1 ω1^2 + \frac{1}{2} I2 ω2^2 + \frac{1}{2} I3 ω3^2$
- Calculation: The kinetic energy of a rotating rigid body depends on its moments of inertia about the principal axes and its angular velocities about those axes.
Gyroscopes: Precession frequency described by $\Omega = \frac{mgR}{Iω}$ .
- Calculation: The precession frequency of a gyroscope depends on the mass (m), gravitational acceleration (g), distance from the pivot to the center of mass (R), moment of inertia (I), and angular velocity (ω).
Euler’s Equations: Equations of motion for a rigid body expressed in terms of principal moments of inertia.