What is the Maximum Height a Rolling Marble Reaches on a Rough Hill?

In summary, the problem asks for the maximum height a solid uniform marble will reach when rolling without slipping up a rough hill with a given initial speed, using the conservation of energy equation. The final height will depend on the initial speed.
  • #1
raiderIV
15
0

Homework Statement


A solid uniform marble is rolling without slipping when it approaches the base of a hill with a speed v0.

Find the maximum height above the base that this marble will reach if the hill is rough enough to prevent slipping.


Homework Equations


Since there is no slipping I can use the conservation of energy

mgh = .5mv2 + .5Iw2

and i end up with the equation

v02 = 10gh/7


The Attempt at a Solution



I have no idea where to go from here, so if someone might be willing to give me a point in the correct direction, it would be most appreciated.
 
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  • #2
raiderIV said:

Homework Statement


A solid uniform marble is rolling without slipping when it approaches the base of a hill with a speed v0.

Find the maximum height above the base that this marble will reach if the hill is rough enough to prevent slipping.


Homework Equations


Since there is no slipping I can use the conservation of energy

mgh = .5mv2 + .5Iw2

and i end up with the equation

v02 = 10gh/7


The Attempt at a Solution



I have no idea where to go from here, so if someone might be willing to give me a point in the correct direction, it would be most appreciated.

You seem pretty much done. You equated the final PE with the initial total KE (linear and rotational), and the final PE has the final height in it that you are asked to solve for. Just re-arrange your final equation.


EDIT -- actually I'm not sure I understand your final equation. You can eliminate the I in the initial equation by using the definition of a sphere's I in terms of its mass and radius. It does seem like your final answer will still have variables in it for m, R and Vo.
 
  • #3
berkeman said:
You seem pretty much done. You equated the final PE with the initial total KE (linear and rotational), and the final PE has the final height in it that you are asked to solve for. Just re-arrange your final equation.EDIT -- actually I'm not sure I understand your final equation. You can eliminate the I in the initial equation by using the definition of a sphere's I in terms of its mass and radius. It does seem like your final answer will still have variables in it for m, R and Vo.
The problem is, that the final solution is asking for a set height in meters. I tried plugging in 9.81 for gravity and solving for h and leaving v0 in there, but to no avail (Its a computer system that i need to input the answers to, and it just tells me that the answer does not use v0 in it). I just don't know if there is someway to find h other than what i am aware of.
 
  • #4
raiderIV said:
The problem is, that the final solution is asking for a set height in meters. I tried plugging in 9.81 for gravity and solving for h and leaving v0 in there, but to no avail (Its a computer system that i need to input the answers to, and it just tells me that the answer does not use v0 in it). I just don't know if there is someway to find h other than what i am aware of.

The final height will depend on Vo. Right? Maybe not on the mass, but for sure on the Vo. What are you not posting about the original problem statement?
 

FAQ: What is the Maximum Height a Rolling Marble Reaches on a Rough Hill?

What is the velocity of a sphere going up a hill?

The velocity of a sphere going up a hill depends on several factors such as the mass of the sphere, the angle of the hill, and the force applied. In general, the velocity will decrease as the sphere moves up the hill due to the force of gravity acting against it.

How is the velocity of a sphere going up a hill affected by its mass?

The velocity of a sphere going up a hill is directly affected by its mass. The greater the mass of the sphere, the harder it is for it to overcome the force of gravity and the slower its velocity will be. This is due to the relationship between mass and inertia, which is the resistance to change in motion.

Can the angle of the hill affect the velocity of a sphere going up?

Yes, the angle of the hill can affect the velocity of a sphere going up. The steeper the hill, the greater the force of gravity pulling the sphere down, which will result in a slower velocity. On the other hand, a less steep hill will have a smaller force of gravity, allowing the sphere to move up at a higher velocity.

What happens to the velocity of a sphere going up a hill if a force is applied?

If a force is applied to a sphere going up a hill, it can increase or decrease its velocity depending on the direction and magnitude of the force. If the force is applied in the same direction as the motion of the sphere, it can increase the velocity. However, if the force is applied in the opposite direction, it can decrease the velocity or even cause the sphere to roll back down the hill.

Is there a maximum velocity a sphere can achieve when going up a hill?

Yes, there is a maximum velocity that a sphere can achieve when going up a hill. This is known as the terminal velocity, which is the highest speed that a falling object can reach when the force of gravity is balanced by the force of air resistance. The terminal velocity will vary depending on the shape, size, and mass of the sphere.

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