The Ball That Would Not Go Further
A steel ball is released from rest and rolls down a smooth ramp, then slides along a rough horizontal wooden floor until it stops. The ball is released from a height of 0.20 m above the floor, measured vertically. In trial 1 a 200 g ball stops after travelling 0.80 m along the floor. In trial 2 a 400 g ball of exactly the same size is released from exactly the same height. A classmate predicts that the heavier ball will travel twice as far, because it stores twice as much energy. Assume the ramp exerts no friction, the ball does not bounce, and the friction force between the ball and the floor is proportional to the ball's weight.
1. Predict the stopping distance of the 400 g ball in trial 2. Justify your prediction using energy, and say precisely where your classmate's reasoning goes wrong.
Read the worked answer
The distance is unchanged: 0.80 m.
At the top the ball holds gravitational potential energy mgh. On the frictionless ramp this all becomes kinetic energy at the bottom, so ½mv² = mgh. On the floor the friction force f = μmg removes that energy over distance d:
μmg · d = mgh → d = h/μ
The mass cancels completely. Doubling the mass does double the energy stored, but it also doubles the friction force that has to remove it, so the ball covers exactly the same distance. The classmate looked at the energy and forgot the retarding force.
2. The floor is then polished, and careful measurement shows the friction force on the ball is halved. Predict the new stopping distance for the 200 g ball, and explain the change in one or two sentences.
Read the worked answer
The new distance is 1.60 m.
Because d = h/μ, and the friction force (and therefore μ) is halved while the release height is unchanged, the ball must travel twice as far to lose the same kinetic energy. The speed at the bottom is unaffected, so nothing else in the situation has changed.
3. Explain why the speed of the ball at the bottom of the ramp does not depend on its mass. Use the idea of acceleration along the ramp in your answer.
Read the worked answer
On the ramp the force pulling the ball along the slope is mg sin θ, and the mass being accelerated is m, so
a = (mg sin θ)/m = g sin θ
The mass cancels again. Every ball released from rest at the same height therefore has the same acceleration and travels the same length of ramp, so it reaches the bottom at the same speed. Mass increases the pulling force and the inertia in exactly the same proportion, which is why it has no effect.