Rotational Motion
Explore angular kinematics, torque, rotational inertia, and angular momentum through interactive simulations.
Rotational motion describes objects spinning around an axis. Just as linear motion has position, velocity, and acceleration, rotational motion has angular position (θ), angular velocity (ω), and angular acceleration (α).
Angular Displacement
where s is arc length and r is radius (measured in radians)
Angular Velocity
Rate of change of angular position (rad/s)
Angular Acceleration
Rate of change of angular velocity (rad/s²)
Rotational Kinematics Equations
Relationship to Linear Motion
where a_t is tangential acceleration and a_c is centripetal acceleration
Torque is the rotational equivalent of force. It causes angular acceleration and depends on both the magnitude of the force and the distance from the axis of rotation (lever arm).
Torque
where r is the distance from axis, F is force, and θ is the angle between them
Newton's Second Law for Rotation
where I is rotational inertia (moment of inertia)
Rotational Inertia
Resistance to rotational acceleration, depends on mass distribution
Common moments of inertia:
• Point mass: I = mr²
• Solid disk/cylinder: I = ½mr²
• Hollow cylinder: I = mr²
• Solid sphere: I = ⅖mr²
• Thin rod (center): I = 1/12 mL²
Rotational Kinetic Energy
Energy of rotation, analogous to ½mv² for linear motion
Torque
7.50 N⋅m
Angular Velocity
0.00 rad/s
Angular Acceleration
0.38 rad/s²
Moment of Inertia
20.00 kg⋅m²
Moment of Inertia
2.500 kg⋅m²
Angular Acceleration
0.80 rad/s²
Angular Velocity
0.00 rad/s
Notice: With the same torque and mass, the disk accelerates fastest (lowest I), the ring accelerates slowest (highest I), and the rod is in between. This demonstrates how mass distribution affects rotational inertia.
Angular momentum is the rotational equivalent of linear momentum. It is conserved in the absence of external torques, leading to fascinating phenomena like ice skaters spinning faster when they pull their arms in.
Angular Momentum
For a point mass: L = mvr sin(θ)
Conservation of Angular Momentum
When no external torque acts on a system
Torque and Angular Momentum
Net torque equals the rate of change of angular momentum
Rolling Motion
Total kinetic energy includes both translational and rotational components
For rolling without slipping: v = rω
Start simulation first
Moment of Inertia
12.25 kg⋅m²
Angular Velocity
0.00 rad/s
Angular Momentum
0.00 kg⋅m²/s
Rotational KE
0.00 J
Conservation of Angular Momentum: Notice that angular momentum (shown in green) stays constant at 24.50 kg⋅m²/s even as you change arm extension. When arms pull in (decreasing I), angular velocity increases proportionally. This is why ice skaters spin faster when they pull their arms in!