Angular displacement (θ):
* θ = Δs / r, where Δs is the arc length traveled and r is the radius of the circular path.
Angular velocity (ω):
* ω = Δθ / Δt, where Δθ is the change in angular displacement and Δt is the time interval.
* ω = 2πf, where f is the frequency of rotation (number of revolutions per second).
Angular acceleration (α):
* α = Δω / Δt, where Δω is the change in angular velocity and Δt is the time interval.
* α = τ / I, where τ is the net torque acting on the object and I is the moment of inertia.
Torque (τ):
* τ = r × F, where r is the distance from the axis of rotation to the point where the force is applied and F is the force.
* τ = Iα, where I is the moment of inertia and α is the angular acceleration.
Moment of inertia (I):
* I = ∑mr², where m is the mass of each particle and r is its distance from the axis of rotation.
* I = 1/2MR², for a solid sphere rotating about its diameter, where M is the mass and R is the radius.
* I = 1/12ML², for a rod rotating about its center, where M is the mass and L is the length.
Kinetic energy of rotation (K_rot):
* K_rot = 1/2Iω², where I is the moment of inertia and ω is the angular velocity.
Work done by a torque (W):
* W = τΔθ, where τ is the torque and Δθ is the angular displacement.
Angular momentum (L):
* L = Iω, where I is the moment of inertia and ω is the angular velocity.
* L = r × p, where r is the position vector and p is the linear momentum.
Conservation of angular momentum:
* If no external torque acts on a system, its total angular momentum remains constant.
These are just some of the most common formulas. There are many others depending on the specific situation and what you want to calculate.
It's important to understand the concepts behind these formulas and how they relate to each other. With practice, you'll be able to apply them confidently to solve problems in rotational motion.
