Physics

Problem

A solid sphere of mass

0.80\,\text{kg}

and radius

0.050\,\text{m}

sits in the back of a truck. The truck’s bed is horizontal and sufficiently rough that the sphere can roll without slipping relative to the truck. The truck begins to accelerate in a straight line along the road. The moment of inertia of a solid sphere about its central axis is

\tfrac25MR^{2}

. The coefficient of static friction between the sphere and the truck bed is

\mu_s=0.40

. (a) Analyze the forces acting on the sphere in an inertial reference frame fixed to Earth. Draw a free-body diagram and identify which force is responsible for the sphere’s horizontal acceleration. (b) Using appropriate equations of motion, derive an inequality that relates the truck’s linear acceleration

a_{\text{truck}}

\mu_s

that must be satisfied to prevent slipping between the sphere and the truck bed. (c) Determine the maximum value of

a_{\text{truck}}

for which rolling without slipping can be maintained. (d) Qualitatively discuss the subsequent motion of the sphere relative to the truck bed if the truck’s acceleration exceeds the value found in part (c).

Intuitive explanation

Rolling without slipping means the sphere's acceleration and angular acceleration are locked together by the constraint $a = \alpha R$ —so friction must do double duty, both accelerating the center of mass and creating the torque needed for rotation. The key insight is that friction has a maximum limit; if the truck accelerates too hard, static friction can't provide enough force to keep the sphere rolling with the truck, and it will slip. Set up the force and torque equations, use the no-slip condition to relate them, and find the critical acceleration where friction reaches its maximum.

Intuitive explanation

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