Physics

When a rigid body rolls without slipping, its motion is constrained—the contact point can't slide, which locks together how fast the center moves and how fast it spins. The key insight is that this no-slip condition creates a geometric relationship: the linear distance the center travels must equal the arc length traced by the rolling surface. Differentiate that constraint twice to connect $a_{\text{cm}}$ and $\alpha$ directly, revealing that they're proportional through the radius.

When a rigid body rolls without slipping, its motion is constrained—the contact point can't slide, which locks together how fast the center moves and how fast it spins. The key insight is that this no-slip condition creates a geometric relationship: the linear distance the center travels must equal the arc length traced by the rolling surface. Differentiate that constraint twice to connect $a_{\text{cm}}$ and $\alpha$ directly, revealing that they're proportional through the radius.