Question

A ball is rolling along at speed v without slipping on a horizontal surface when it comes to a hill that rises at a constant angle above the horizontal. In which case will it go higher up the hill: if the hill has enough friction to prevent slipping, or if the hill is perfectly smooth. Justify your answer with a conservation of energy statement

Answer 1

The ball will go higher up the hill if the hill has enough friction to prevent slipping.

Step-by-step explanation:

The ball will go higher up the hill if the hill has enough friction to prevent slipping. This is because in the case where there is enough friction, the ball can convert some of its kinetic energy to rotational energy, allowing it to roll up the hill. The conservation of energy statement can be used to explain this:

When the ball rolls without slipping, its total mechanical energy is conserved.

As the ball rolls up the hill, its potential energy increases and its kinetic energy decreases.

In the case where there is enough friction, some of the kinetic energy is converted to rotational energy, allowing the ball to reach a higher height on the hill.

Therefore, the ball will go higher up the hill if the hill has enough friction to prevent slipping.

Answer 2

The ball has an initial linear kinetic energy and initial rotational kinetic energy which can both be converted into gravitational potential energy. Therefore the hill with friction will let the ball reach higher.

Step-by-step explanation:

The ball has an initial linear kinetic energy and initial rotational kinetic energy which can both be converted into gravitational potential energy. Therefore the hill with friction will let the ball reach higher.

This is because:

If we consider the ball initially at rest on a frictionless surface and a force is exerted through the centre of mass of the ball, it will slide across the surface with no rotation, and thus, there will only be translational motion.

Now, if there is friction and force is again applied to the stationary ball, the frictional force will act in the opposite direction to the force but at the edge of the ball that rests on the ground. This friction generates a torque on the ball which starts the rotation.

Therefore, static friction is infact necessary for a ball to begin rolling.

Now, from the top of the ball, it will move at a speed 2v, while the centre of mass of the ball will move at a speed v and lastly, the bottom edge of the ball will instantaneously be at rest. So as the edge touching the ground is stationary, it experiences no friction.

So friction is necessary for a ball to start rolling but once the rolling condition has been met the ball experiences no friction.