# Ball goes down a frictionless ramp

• M1ZeN
In summary: Also, the author of the post says "no rolling" in his very first postPretty sure the author of the post was referring to "no rolling" in the first part of the problem, as in the ball is not rolling but rather sliding down the ramp.In summary, the problem deals with a frictionless ramp and a hollow ball with a given mass and radius. The objective is to find the final velocity of the ball at the bottom of the ramp if it slides down to a final height of 0 m, assuming no rolling. The solution involves using the principle of mechanical energy conservation, and ultimately substituting the rotational kinetic energy formula to solve for the final velocity. Additionally, the problem asks what the angular velocity of the ball would be
M1ZeN

## Homework Statement

A hollow ball goes down a frictionless ramp. The ball starts at a height of 5 m. The ball has a radius of 4 cm and a mass of 0.5 kg.

(a) What is the final velocity of the ball at the ball at the bottom of the ramp if it just slides down the ramp to a final height of 0 m. (Assume no rolling)

(b) If the work done (by gravity on the ball) was used to rotate the sphere instead, what would the angular velocity of the ball be?

## Homework Equations

I have no clue. I went through the section in my textbook that seems to relate to this practice test problem but they none of them seems to be anything I could use.

## The Attempt at a Solution

Again, being I can't find any formulas or equations to work with. I don't know where to start with the values given.

Thanks

Last edited:
Hi there,

Since your questions asked for the "work done", I presume that you are studying the chapter dealing with energy. Remember that a frictionless problem boils down to the mechanical energy conservation principle.

Cheers

I see. That definitely rang a bell.

So simply, I could just break the problem down with K(i) + U(i) = K(f) + U(f) and in turn that would substitute to be (1/2)MVi^2 + mgyi = (1/2)MVf^2 + mgyf

Also, for part b, would I just use for angular velocity w = sqrt(g/r)?

No; I think what you need to use for b is conservation of energy where there are 2 forms of kinetic energy, rotational and translational:

mgy=1/2mv^2+1/2(I)w^2 So you need to compute I for a hollow ball (I=2/5mr^2).

Now altho it may look hopeless, one eqn with 2 unknowns it is not. The translational velocity (linear) and rotational or angular velocity are related how? Use this to substitute v with w (angular velocity).

M1ZeN said:
(a) What is the final velocity of the ball at the ball at the bottom of the ramp if it just slides down the ramp to a final height of 0 m. (Assume no rolling)

(b) If the work done (by gravity on the ball) was used to rotate the sphere instead, what would the angular velocity of the ball be?
The work done by gravity--and thus the final kinetic energy--in both cases in the same. In the first, all the energy is translational KE; in the second, rotational KE. What's the formula for rotational KE?

I interpret part (b) a bit differently than denverdoc did. I don't think it's asking for the rotational speed of a ball rolling down the hill (which is a more interesting problem). (If that is what it's asking, then it's very poorly worded!)

Doc, I see your point. I thought it was ambiguous and decided on the more "interesting" interpretation. Your likely right as it does say "convert."

## 1. How does the angle of the ramp affect the ball's speed?

The steeper the angle of the ramp, the faster the ball will roll down due to the increased force of gravity pulling it down.

## 2. Why does a ball roll down a frictionless ramp?

The ball rolls down the ramp due to the force of gravity acting on it. In the absence of friction, there is no force to resist the ball's motion, allowing it to roll down the ramp with increasing speed.

## 3. What would happen if there was friction on the ramp?

If there was friction on the ramp, it would act against the ball's motion, causing it to slow down and potentially stop before reaching the bottom of the ramp. Friction can also cause the ball to roll at a slower and more constant speed.

## 4. Does the mass of the ball affect its speed down the ramp?

Yes, the mass of the ball does affect its speed down the ramp. A heavier ball will have more inertia, meaning it will resist changes in motion. Therefore, a heavier ball will roll down the ramp slower than a lighter ball due to the force of gravity being more evenly distributed across its mass.

## 5. How does the length of the ramp impact the ball's speed?

The longer the ramp, the longer the distance the ball has to accelerate before reaching the bottom. This means that the ball will have a higher speed at the bottom of a longer ramp compared to a shorter ramp. However, the angle of the ramp also plays a significant role in the ball's speed, so a shorter ramp with a steeper angle can still result in a faster ball speed than a longer ramp with a shallower angle.

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