The energy an object possesses due to its rotation.

How resistant an object is to changes in its rotation.

How fast the object is rotating, measured in radians per second.

The kinetic energy an object possesses due to its linear motion.

Rotational kinetic energy changes when work is done by a torque.

Rotational: Energy due to rotation, depends on moment of inertia and angular velocity. Translational: Energy due to linear motion, depends on mass and linear velocity.

Scalar: Magnitude only (e.g., rotational kinetic energy). Vector: Magnitude and direction (e.g., angular velocity, angular momentum).

Solid Sphere: $I = (2/5)MR^2$ (mass concentrated closer to the center). Hollow Sphere: $I = (2/3)MR^2$ (mass distributed further from the center).

Rolling without slipping: $v = Rω$ (linear and angular velocities are related). Rolling with slipping: $v ≠ Rω$ (linear and angular velocities are independent).

Angular Momentum: Vector quantity, measures the quantity of rotation. Rotational Kinetic Energy: Scalar quantity, measures the energy of rotation.

1. Calculate rotational kinetic energy: $K_{rot} = (1/2)Iω^2$. 2. Calculate translational kinetic energy: $K_{trans} = (1/2)mv^2$. 3. Add them together: $K_{total} = K_{rot} + K_{trans}$.

1. Identify knowns and unknowns. 2. Choose the appropriate formula. 3. Convert units to radians per second if necessary. 4. Substitute values and solve.

1. Identify initial and final states. 2. Determine potential and kinetic energies at each state. 3. Apply the conservation of energy equation: $E_{initial} = E_{final}$. 4. Solve for the unknown variable.

1. Determine the rotational inertia (I) of the system about its center of mass. 2. Measure or calculate the angular velocity (ω) of the system. 3. Apply the formula: $K_{rot} = (1/2)Iω^2$.

1. Identify the radius (R) of the rolling object. 2. Determine the angular velocity (ω) in radians per second. 3. Use the equation: $v = Rω$ to find the linear velocity (v).