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Q & A: balls rolling down the ramp

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Most recent answer: 10/22/2007
Q:
If a ball is running down a ramp, why is it that when you change the height of the ramp, the ball runs down the ramp faster?
- Anonymous
A:
If you increase the steepness of the ramp, then you will increase the acceleration of a ball which rolls down the ramp. This can be seen in two different ways:

1) Components of forces. Forces are vectors and have a direction and a magnitude. The force of gravity points straight down, but a ball rolling down a ramp doesn't go straight down, it follows the ramp. Therefore, only the component of the gravitational force which points along the direction of the ball's motion can accelerate the ball. The other component pushes the ball into the ramp, and the ramp pushes back, so there is no acceleration of the ball into the ramp. If the ramp is horizontal, then the ball does not accelerate, as gravity pushes the ball into the ramp and not along the surface of the ramp. If the ramp is vertical, the ball just drops with acceleration due to gravity. These arguments are changed a bit by the fact that the ball is rolling and not sliding, but that only affects the magnitude of the acceleration but not the fact that it increases with ramp steepness.

2) Work and energy. The change in potential energy of the ball is its mass times the change in height (only the vertical component counts -- horizontal displacements do not change gravitational potential energy) times the local gravitational acceleration g. This loss of gravitational potential energy shows up as an increase in kinetic energy. If the ball falls a farther distance vertically, it will have a greater kinetic energy and be going faster. Again, the kinetic energy is shared between the motion of the ball going somewhere, and the rotation of the ball, and so the details of the acceleration depend on the ball (is it hollow or solid?), but the dependence on the steepness of the ramp is the same.

Tom

(published on 10/22/2007)

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