How Is Rolling Without Slipping Analyzed?

Rolling without slipping is analyzed as simultaneous translational and rotational motion linked by the condition $v_{\mathrm{CM}}=\omega R$. This is an HL-only application of A.4 Rigid body mechanics.

The Rolling Condition

For a wheel of radius $R$, the distance moved by its centre of mass equals the arc length rotated:

$$s=R\theta$$

Differentiating gives $v_{\mathrm{CM}}=\omega R$, and differentiating again gives $a_{\mathrm{CM}}=\alpha R$. These conditions apply only when there is no relative motion between the wheel and the surface at the instantaneous contact point.

The contact point is instantaneously at rest relative to the ground. The centre of mass still moves forward, while the wheel rotates around it.

Methods of Analysis

Use the force-and-torque method when friction, acceleration, or the minimum coefficient of static friction is required. Choose the centre of mass as the torque axis so the weight and normal force often produce no torque.

For an object rolling down an incline at angle $\theta$, resolve weight parallel to the slope, write equations for translation and rotation, then apply the rolling condition. Finally, verify that $F_f\leq\mu_sN$; otherwise the required static friction is unavailable and slipping occurs.

The friction involved is normally static friction, because the surfaces do not slide relative to each other. Static friction supplies any required torque, but it is not automatically equal to $\mu_sN$; instead, $F_f\leq\mu_sN$.

For example, a cylinder with $I=\frac{1}{2}MR^2$ rolls from rest through a vertical height $h$. Conservation of energy gives

$$Mgh=\frac{1}{2}Mv^2+\frac{1}{2}\left(\frac{1}{2}MR^2\right)\left(\frac{v}{R}\right)^2.$$

Therefore, $Mgh=\frac{3}{4}Mv^2$, so

$$v=\sqrt{\frac{4gh}{3}}.$$

A common misconception is that friction must always oppose the centre-of-mass motion. In rolling, static friction opposes the tendency to slip, so its direction depends on the forces and torques acting.

Exam Technique

In an IB response, state the rolling condition before combining the translational and rotational equations. Keep $M$, $R$, $I$, $v$, and $\omega$ clearly identified, and do not omit rotational kinetic energy.