Summary
Highlights
Translational kinetic energy is 1/2 * mass * speed squared. Rotational kinetic energy is 1/2 * rotational inertia * angular speed squared. Rotational inertia is the rotational equivalent of mass, and angular speed is the rotational equivalent of speed. The total kinetic energy of a rigid object is the sum of its rotational kinetic energy about its center of mass and its translational kinetic energy.
Work done by a force is force * displacement * cos(theta). Work done by a torque is torque * angular displacement. In AP Physics 1, torque and angular displacement are either clockwise or counterclockwise, so the cosine term is not needed, but the sign of the work depends on their relative directions. On a graph of force vs. position, work is the signed area under the curve; similarly, on a graph of torque vs. angular position, work is the signed area.
Linear momentum (p) equals mass * velocity. Angular momentum (L) equals rotational inertia * angular velocity for a rigid object with shape relative to an axis of rotation. For a point particle, angular momentum is r * m * v * sin(theta). Both linear and angular momentum are vectors. In AP Physics 1, clockwise and counterclockwise indicate the direction of angular momentum. Units for angular momentum are kg*m²/s.
Newton's Second Law states net force = mass * acceleration or net force = change in momentum / change in time. The rotational forms are net torque = rotational inertia * angular acceleration or net torque = change in angular momentum / change in time. Angular impulse equals change in angular momentum, which also equals impact torque * change in time, or the area under an impact torque versus time curve. The slope of an angular momentum vs. time graph is the net torque.
If the net torque on a system is zero, then the change in angular momentum is zero, meaning initial angular momentum equals final angular momentum. This is demonstrated by an ice skater pulling in their arms: decreasing rotational inertia leads to an increase in angular velocity because angular momentum is conserved if no external torque is present. Newton's Third Law also has a rotational form: torque by object 1 on object 2 is equal and opposite to torque by object 2 on object 1, leading to equal and opposite angular impulses.
For an object rolling without slipping, the displacement, velocity, and acceleration of its center of mass are related to angular displacement, angular velocity, and angular acceleration by the radius of the object (e.g., displacement = R * angular displacement). The uppercase R denotes the constant radius of the object. An object rolling without slipping has both translational and rotational kinetic energies. In an ideal case, static friction does no work. The acceleration of an object rolling down an incline depends on the incline angle, gravitational field, and the factor in front of mass*radius squared in the rotational inertia equation. Objects with more mass distributed closer to their center of mass will accelerate faster down an incline.
When an object rolls with slipping, the equations for rolling without slipping are no longer valid, and kinetic friction dissipates energy by doing work. For orbiting satellites, in circular orbits, total mechanical energy, gravitational potential energy, satellite's angular momentum, and kinetic energy remain constant. In elliptical orbits, total mechanical energy and satellite's angular momentum remain constant, but gravitational potential energy and kinetic energy do not. A satellite's linear momentum does not remain constant due to changing velocity direction, but its angular momentum does, as the net torque from gravity about the planet's center is zero. Escape velocity is the minimum speed an object needs to reach an infinite distance from a planet with zero final velocity, ignoring frictional effects.