Inertia of a hollow cylinder and hollow sphere

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Homework Help Overview

The discussion revolves around calculating the inner radius of a hollow cylinder that rolls down an incline in the same time as a hollow sphere, both having the same mass and outer radius. The problem involves concepts of moment of inertia and the dynamics of rolling objects.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants explore the moment of inertia for both the hollow cylinder and the hollow sphere, with some questioning the initial equations provided for the inertia of a hollow cylinder. There are attempts to derive the correct formula for the inertia of a hollow cylinder with finite thickness, and discussions about the implications of varying the inner radius.

Discussion Status

There is ongoing exploration of the correct expressions for moment of inertia, with some participants suggesting reconsideration of the density of the cylinder. Multiple interpretations of the inertia calculations are being discussed, and some participants appear to be converging on a formula, though consensus has not been reached.

Contextual Notes

Participants note the complexity of calculating the inertia due to the hollow nature of the cylinder and the need to consider its density. There is also an emphasis on the relationship between the inner and outer radii in the context of the problem.

GayYoda
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Homework Statement


Consider a hollow cylinder of mass M with an outer radius R_out = 10 cm and an unknown inner radius R_in. If the hollow cylinder is to roll down an incline in the same time as a spherical shell of the same mass and the same outer radius, calculate R_in.

Homework Equations


I_cyl = MR^2/2 .
I_shell = 3/5(MR^2)

The Attempt at a Solution


I think the inertia of each are equal but I'm not sure
 
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You've given the moment of inertia of a solid cylinder. What you need is the inertia of a hollow cylinder of finite thickness.
 
I=M/2[(R_out)^2-(R_in)^2]?
 
GayYoda said:
I=M/2[(R_out)^2-(R_in)^2]?
What happens in your formula if ##R_{in}## is close to ##R_{out}##?

To calculate this inertia is not that easy. I suggest you consider the density of the cylinder.
 
i got it now its I=M[(R_out)^2+(R_in)^2]/2 as the density becomes M/[pi*h*[(R_out)^2-(R_in)^2] and when you sub it back into the intergral it becomes [(R_out)^4-(R_in)^4]/[(R_out)^2-(R_in)^2] = [(R_out)^2+(R_in)^2] because of difference of 2 squares
 

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