Rotation and Systems
(Level 3)
Levels 1 and 2 covered linear motion and forces. Level 3 asks: what happens when things rotate? Most machines have rotating parts. Shafts spin, gears mesh, motors turn, wheels roll. Forces applied at a distance cause rotation. Multiple parts move together as systems.
This level gives you the physics to analyze anything that rotates or involves multiple interacting components.
Angular Motion
Rotation is just linear motion wrapped around a circle. Same physics, different coordinates. Position becomes angle θ. Velocity becomes angular velocity ω (radians per second). Acceleration becomes angular acceleration α.
| Linear | Rotational |
|---|---|
| Position x | Angle θ |
| Velocity v | Angular velocity ω |
| Acceleration a | Angular acceleration α |
Same kinematic equations work. Constant angular acceleration? ω = ω₀ + αt. Traveled angle? θ = θ₀ + ω₀t + ½αt². Replace linear variables with angular ones and you're done.
After 5 seconds: ω = 0 + (2)(5) = 10 rad/s
Angle rotated: θ = ½(2)(5)² = 25 radians ≈ 4 full rotations
Motors, gears, turbines, wheels—anything that spins uses these equations. Once you know linear kinematics, rotational kinematics is free.
Torque
Force causes linear acceleration. Torque causes angular acceleration. Push on a door handle and it rotates. Push near the hinge? Barely moves. Same force, different torque. Torque depends on both force magnitude and where you apply it.
r is the distance from the rotation axis to where the force acts. F is the applied force. The cross product means only the perpendicular component of force contributes. Push radially inward (toward the axis)? Zero torque. Push tangentially (perpendicular to r)? Maximum torque.
Force = 50 N, wrench length = 0.3 m, perpendicular pull
Pull at an angle? Multiply by sin θ to get the perpendicular component. Pull parallel to the wrench? τ = 0, you're just pushing/pulling the bolt, not turning it.
Torque shows up everywhere: bolted joints, motor shafts, steering systems, structural loading. Design specs give you torque limits, not just force limits. 100 N·m on a fastener means something precise—force alone doesn't tell the whole story.
Newton's Second Law for Rotation
Linear motion: F = ma. Rotational motion: Στ = Iα. Net torque equals moment of inertia times angular acceleration. Same structure, rotational variables.
What's I? Moment of inertia. It's the rotational equivalent of mass. Mass resists linear acceleration. Moment of inertia resists angular acceleration. But unlike mass (which is just a number), moment of inertia depends on how that mass is distributed around the axis.
Mass farther from the axis → larger I → harder to spin up or slow down.
Solid disk rotating about center: I = ½mr²
Solid sphere: I = ⅖mr²
Thin rod about center: I = 1/12 mL²
You don't memorize these. Look them up when needed. What matters: understand that shape and axis location change I dramatically.
Two objects, same mass. Compact sphere vs hollow ring. Ring has more mass far from the axis. Higher I. Apply same torque, ring accelerates slower. This is why flywheels are designed with mass at the rim—you want high I to store rotational energy and resist speed changes.
Moment of inertia affects everything: vibration frequencies, response time, structural dynamics. Change the geometry, you change the behavior. It's not just about how heavy something is—it's about where that mass sits.
Angular Momentum
Linear momentum: p = mv. Angular momentum: L = Iω. If net torque is zero, angular momentum stays constant. Just like linear momentum conservation, but for rotation.
No external torque → L = Iω = constant
Arms out: large I, slow ω
Arms in: small I, fast ω
I decreases, so ω increases to keep L constant. Spin faster without applying any external force.
This shows up in machinery, gyroscopes, spacecraft attitude control, vehicle dynamics. Change the mass distribution while spinning and the rotation rate adjusts automatically. Angular momentum conservation is the reason spinning objects are stable—external torque is required to change their rotational state.
Rigid-Body Systems
Real objects deform when loaded. Rigid body assumption: treat the object as if it doesn't. Reasonable approximation when deformation is small compared to overall motion. Lets you analyze rotation and translation together without worrying about internal stresses (yet).
Single particle? One equation: F = ma. Rigid body? Need two: ΣF = ma (for translation of center of mass) and Στ = Iα (for rotation about center of mass). Forces act at different points. Some cause translation, some cause rotation, most cause both.
Force equation tells you the center of mass acceleration.
Torque equation tells you the angular acceleration about the support.
Both happen simultaneously. You need both equations to fully describe motion.
Most mechanical systems are multi-body: linkages, gears, vehicles, robots. Each part has its own mass, inertia, constraints. Motions are coupled—one part moving affects others. You end up with systems of equations, not single formulas. This is where dynamics gets complex, and also where it gets real. Mechanism design, multibody simulation, robotics—all built on rigid-body analysis.
Task: Rotational Motion Under Applied Torque (With Full Worked Answer)
A solid disk of mass m = 10 kg and radius r = 0.5 m rotates about a fixed axis through its center. A constant torque of 20 N·m is applied.
1. Find the angular acceleration of the disk.
2. Find the angular velocity after 4 s, assuming it starts from rest.
3. Explain why mass distribution matters more than total mass.
Solution
For a solid disk:
Substitute values:
Answer (1): α = 16 rad/s²
Since it starts from rest:
Answer (2): ω = 64 rad/s
Even with the same mass, moving mass farther from the axis increases moment of inertia. A larger I means smaller angular acceleration for the same torque.
Answer (3): Rotation depends on how mass is distributed relative to the axis, not just how much mass exists.
What You Should Take From Physics Level 3
- Rotation is governed by torque, not force alone
- Angular motion mirrors linear motion mathematically
- Moment of inertia controls system response
- Systems must be analyzed as interacting parts
Ready for the Next Level?
Once you understand rotation and multi-body systems, you're ready to explore oscillatory motion and wave phenomena.
Continue to Level 4: Vibrations and Waves →