1.7. Rigid-Body Mechanics (A4 - HL Only)

HL Only

Many linear motion concepts have rotational analogs, forming a parallel structure
1. Displacement s, Angular displacement θ
2. Velocity v, Angular velocity ω
3. Acceleration a, Angular acceleration α
4. Mass m, Moment of inertia I
5. Force F, Torque τ
6. F=ma, τ=Iα
This parallel allows us to apply familiar motion principles (like Newton’s Laws) to rotating systems.

Torque is the rotational equivalent of force
Just like net force causes linear acceleration, net torque causes angular acceleration
By analyzing the torques acting on an object, we can predict how fast it will spin up, slow down, or maintain rotational equilibrium.
If ∑τ=0, the system is in rotational equilibrium (either at rest or rotating at constant ω).

This is all about moment of inertia I. It depends not just on mass, but how far that mass is from the axis of rotation.
A body with more mass farther from the axis has a larger moment of inertia, making it harder to spin or stop spinning. Example: A solid disk vs. a ring of the same mass and radius — the ring has higher III, so it resists changes in rotation more.
In applications like figure skating or gyroscopes, shifting mass inward or outward changes I, which affects angular velocity due to conservation of angular momentum: L = Iω = constant

the torque of a force about an axis
that bodies in rotational equilibrium have a resultant torque of zero
that an unbalanced torque applied to an extended, rigid body will cause angular acceleration
that the rotation of a body can be described in terms of angular displacement, angular velocity and angular acceleration
that equations of motion for uniform angular acceleration can be used to predict the body's angular position, angular displacement, angular speed and angular acceleration
that the moment of inertia I depends on the distribution of mass of an extended body about an axis of rotation
the moment of inertia for a system of point masses
Newton's second law for rotation where T is the average torque that an extended body rotating with an angular speed has an angular momentum L
that angular momentum remains constant unless the body is acted upon by a resultant torque
that the action of a resultant torque constitutes an angular impulse L
the kinetic energy of rotational motion