• A combo of translational (straight line) and rotational (spinning) motion.
• Key formula: Total kinetic energy (K.E) = \(\frac 12mu^2 + \frac 12 Iw^2\)

• When an object rolls down a height, ΔℎΔh, it loses potential energy (P.E) equal to mgΔh.
• Conservation of energy:2mgΔh = \(\frac 12\)​ mv² + \(\frac 12\) ​Iω².

🌍 Real-world example: Think of a bowling ball and a marble. Even if they start rolling down a slope at the same time, they might not reach the bottom simultaneously. This is due to their different shapes and distribution of mass!

• Rolling on an Incline
• Forces on the rolling object: weight (Mg), normal force (N), and static friction (Ff).
  • Friction causes the rotation.
  • Linear acceleration, a is related to angular acceleration, α, as a=αR.

Shapes & their accelerations

Solid sphere
a = \(\frac 57\)​ gsinθ.

ii. Solid cylinder: a = \(\frac 23\) gsinθ.

iii. Thin hoop: a = \(\frac 12\) gsinθ.

Result: Spheres go zoom-zoom the fastest, hoops laze around!

• Which has more rotational K.E?
• The sphere has higher speed and more translational K.E than a cylinder.
• BUT! The cylinder spins more and has a higher rotational K.E.

Car wheels spin as the car moves. If the car is a Ferrari moving at 20 m/s, the wheels have both translational K.E (due to the car's motion) and rotational K.E (from spinning).