Cylinder rolling down an inclined plane

[trisectedangl]

This is an even-numbered problem in my textbook that I'm looking at early though it hasn't been taught by my lecturer yet, because I need to know the concepts underlying it for a similar problem. Cld someone help me?

Homework Statement

Consider the case of a hollow cylinder rolling down a plane inclined to the ground at an angle β. There is a small piece of plasticine stuck to a fixed position on the cylinder's inner circumference. What is the acceleration in this case?

Assume the small piece of plasticine to be a rigid cylinder of radius r and mass m, and the hollow cylinder to be of radius R and mass M.

Homework Equations

The formula for acceleration of hollow and filled cylinders

The Attempt at a Solution

Can't solve...


Help please?


Thanks!

 

[shoeman]

i'm not too sure but I'm throwing it out there.

(whenever i type "I" it's for moment of inertia, when talking about myself i'll use lower case i).
i assume you need first to find the moment of inertia through the center of the body, then use Steiner's equation to find I on the edge that is touching the incline.
The hardest part here is that I of the edge that is touching the incline changes according the where the plasticine is, so you would have to find I as a function of θ (angular position of the plasticine).
and θ is a function of how the body rolls.
if it's rolling without slipping you can do a moment equation on the edge of the body
Ʃτ=Iα
α=a/R
and maybe throw in a force equation or two, i think you should reach a good equation with θ and it's derivatives...
hope this helped.

 

[trisectedangl]

Thanks. But i know these already.

By the way, there is friction, which has nonzero torque on the centre of the can.

 

[trisectedangl]

Could you be more specific and show how you did the maths?

Sorry I'm so amateur-ish

 

[asteriatic]

I understand your need to understand the underlying concepts before they are taught in class. In this problem, there are a few key concepts to consider. First, the acceleration of the hollow cylinder will depend on both its mass and the distribution of that mass. This is because the acceleration is affected by both the force of gravity and the moment of inertia of the cylinder. The moment of inertia is a measure of how the mass is distributed around the axis of rotation, and it can vary depending on the shape and size of the object.

In this case, the small piece of plasticine will also contribute to the moment of inertia of the hollow cylinder. This is because it is stuck to the inner circumference and will rotate with the cylinder. To solve this problem, you will need to use the formula for the moment of inertia of a hollow cylinder and add the contribution from the small piece of plasticine.

Additionally, the angle of the incline will also affect the acceleration of the cylinder. This is because the force of gravity acting on the cylinder will be resolved into components parallel and perpendicular to the incline. The parallel component will cause the cylinder to accelerate down the incline, while the perpendicular component will cause it to rotate.

I would suggest reviewing the concepts of moment of inertia and resolving forces on an inclined plane before attempting to solve this problem. Once you have a solid understanding of these concepts, you can use the given information and equations to solve for the acceleration of the hollow cylinder. Don't be afraid to ask your lecturer or classmates for help as well. Good luck!