Rotation
Rigid body mechanics: angular kinematics, moment of inertia, torque, and rolling.
General idea of the topic
Rotation studies the motion of rigid bodies about an axis. Quantities analogous to those of translation appear: angle, angular velocity and acceleration, the moment of a force, moment of inertia, rotational kinetic energy and angular momentum.
The block follows that parallelism: first the rotation is described, then what causes it is explained, then its energy is treated, then the special case of rolling and, finally, the quantity that is conserved when there are no external moments, angular momentum.
What you will truly understand
By the end you will be able to describe a rotation with angular quantities and convert to linear ones, compute moments and moments of inertia, apply the second law of rotation and use energy and angular momentum when appropriate.
You will also be able to recognize when a body rolls without slipping and what constraint that imposes.
Recommended learning route
- Rotation Fundamentals: angle, angular velocity and acceleration, and their relation to linear quantities.
- Rotational Dynamics: moment of a force, moment of inertia and the second law of rotation.
- Energy in Rotation: rotational kinetic energy, work of the moment and energy in rolling.
- Rolling: rolling with and without slipping and its constraint between translation and spin.
- Angular Momentum: definition, conservation and effects of angular momentum, such as precession.
Physical criterion you must acquire
The essential criterion is to choose the tool according to what is asked: moment equations if the angular acceleration matters, energy if the initial and final states matter without time detail, and angular momentum if there are no external moments. All must give the same result when applied correctly.
Connection with real problems
- wheels, gears and turbines.
- skating, gymnastics and dance.
- gyroscopes, gyrocompasses and satellites.
- flywheels and energy storage.
Final goal of the topic
The final goal is to analyze any rotational motion of a rigid body with confidence and choose the most effective formulation for each problem.
Topics in this section
- Rotation Fundamentals
Fundamental angular variables, vector angular velocity, and baseline moment of inertia.
- Rotational Dynamics (Rotation)
Fundamental equation of rotational dynamics and angular response under external torques.
- Energy in Rotation
Rotational kinetic energy in rigid bodies and mechanical energy conservation.
- Three-Dimensional Rigid Body
Tensorial dynamics of the rigid body. Covers: Euler Equations for a Rigid Body.
- Rolling
Rolling without slipping as a superposition of translational and rotational motion.
- Angular Momentum (Rotation)
Vector angular momentum, conservation laws, and gyroscopic precession in spinning bodies.