Cylinder rolling down a slope

In summary, the problem involves a cylindrical hoop rolling without slipping on a rough incline, then continuing up a smooth incline. The question is what height the hoop will reach. The relevant equations are gravitational potential energy, rotational kinetic energy, linear kinetic energy, moment of inertia, and friction force. The attempt at a solution involves using the conservation of energy equation and substituting in the equations for rotational and linear kinetic energy, resulting in the conclusion that only half of the hoop's energy is available for climbing the incline due to dissipation of rotational energy. The explanation is considered to be somewhat unclear and fudged.
  • #1
nathangrand
40
0
1. Homework Statement

A cylindrical hoop rests on a rough uniform incline. It is released and rolls
without slipping through a vertical distance h0. It then continues up a perfectly
smooth incline. What height does it reach?

2. Homework Equations

GPE=mgh
Rotational Ke =0.5Iw2
Linear Ke = 0.5mv2
Moment of inertia of the hoop, I = mr2 where r is the radius, and m the mass
Friction Force = UN where U is the coefficient of friction, N the normal reaction force

3. The Attempt at a Solution :
Ok..I'm sort of thinking now that at bottom of the slope, mgh0=Ke[Rot] +Ke[Lin]
mgh0={{0.5Iw2}}+0.5mv2
mgh0= {{0.5mv2}} + 0.5mv2 by using v=wr and equation for I

So half the energy is as rotational kinetic energy?
This rotational energy will have to be dissipated as the hoop rolls up the other slope and comes to rest
meaning that only half the energy -- the translational half -- is available for climbing the slope
and hence why the hoop only reaches half the height it started at (I have the answers!)

Can't help but thinking this is a bit of a rubbish explanation and very fudged..can someone tell me if my thinking is right and explain?


Any help appreciated as always guys :)
 
Last edited:
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  • #2
sorted :) would delete the post but not sure how
 

1. How does the mass of the cylinder affect its rolling speed down a slope?

The mass of the cylinder does not affect its rolling speed down a slope. According to the law of inertia, the mass of an object does not impact its acceleration due to gravity. Therefore, a heavier cylinder and a lighter cylinder will have the same rolling speed down a slope.

2. What is the role of the shape of the cylinder in its motion down a slope?

The shape of the cylinder plays a crucial role in its motion down a slope. A cylinder with a larger diameter will have a larger moment of inertia, making it more resistant to rotational motion. As a result, it will roll slower down a slope compared to a cylinder with a smaller diameter.

3. How does the angle of the slope affect the rolling speed of the cylinder?

The steeper the slope, the faster the cylinder will roll down due to an increase in the component of the force of gravity acting parallel to the slope. This can be explained by the equation F = mg sinθ, where θ is the angle of the slope. Therefore, a steeper slope will result in a higher acceleration and faster rolling speed for the cylinder.

4. What is the relationship between the height of the slope and the final velocity of the cylinder?

The height of the slope does not directly affect the final velocity of the cylinder. However, a higher slope will result in a longer distance traveled by the cylinder, which in turn will result in a higher final velocity due to a longer acceleration time. This can be explained by the equation v = u + at, where v is the final velocity, u is the initial velocity (which is zero in this case), a is the acceleration, and t is the time taken to reach the bottom of the slope.

5. Is there a maximum speed that a cylinder can reach when rolling down a slope?

According to the law of conservation of energy, the total mechanical energy (kinetic energy + potential energy) of the cylinder will remain constant throughout its motion down the slope, neglecting any external forces like friction. Therefore, the cylinder will continue to gain speed until it reaches the bottom of the slope, where its potential energy will be zero and its kinetic energy will be at its maximum. In theory, there is no maximum speed for the cylinder when rolling down a slope.

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