Note: These notes cover the fundamental concepts of rotational dynamics, including angular motion, torque, moment of inertia, and more.
1. Basic Concepts
Rotation is the motion of an object around a fixed axis. A rigid body is a body with a perfectly definite and unchanging shape.
2. Angular Displacement, Velocity, and Acceleration
- Angular Displacement (θ): The angle through which a body rotates. Measured in radians (rad).
- Angular Velocity (ω): The rate of change of angular displacement. ω = dθ/dt.
- Angular Acceleration (α): The rate of change of angular velocity. α = dω/dt.
3. Torque and Moment of Inertia
Torque (Ï„): The rotational equivalent of force. Ï„ = r × F.
Moment of Inertia (I): The rotational equivalent of mass. I = ∑máµ¢ráµ¢².
4. Rotational Kinematics
The equations of rotational motion are analogous to linear kinematics:
- θ = θ₀ + ω₀t + ½Î±t²
- ω = ω₀ + αt
- ω² = ω₀² + 2α(θ - θ₀)
5. Rotational Dynamics
Newton's Second Law for Rotation: τ = Iα.
Rotational Kinetic Energy: KE = ½Iω².
6. Angular Momentum
Angular Momentum (L): L = Iω.
Conservation of Angular Momentum: If no external torque acts, Linitial = Lfinal.
7. Rolling Motion
Rolling Without Slipping: Combines translational and rotational motion. v = rω.
8. Parallel Axis Theorem
The moment of inertia about any axis parallel to and a distance d away from the center of mass is: I = Icm + md².
9. Perpendicular Axis Theorem
For a planar object: Iz = Ix + Iy.
10. Applications
- Gyroscopes: Use angular momentum to maintain orientation.
- Flywheels: Store rotational energy.
Key Points to Remember
- Angular velocity and acceleration follow the right-hand rule.
- Torque is a vector quantity perpendicular to the plane of r and F.
- Moment of inertia depends on mass distribution and axis of rotation.
- Conservation of angular momentum is a fundamental principle.
