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Rolling without slipping

When something rolls without slipping, the kinetic energy is split between translation and rotation, and the angular velocity is linked to the speed through the radius. Two equally heavy bodies with different mass distributions can therefore reach different speeds at the bottom of an incline, even starting together.

v = \omega R

relation between speed and angular velocity when rolling

E_k = \tfrac12mv^2 + \tfrac12I\omega^2

total kinetic energy while rolling

Symbols

$v$	speed of the center of mass	m/s
$\omega$	angular velocity	rad/s
$R$	radius	m

Example

A solid cylinder ( $I = \tfrac12mR^2$ ) rolls down $h = 1.5$ m. Energy conservation gives $mgh = \tfrac34mv^2$ , so $v = \sqrt{4\cdot 9.81\cdot 1.5/3} \approx 4.43$ m/s.

A block sliding without friction reaches a higher speed than something rolling, because rolling spends some energy on rotation.

Practise energy and rotation for free →

← Torque and angular momentum

Part of Physics: Energy and rotation.