Why Does a Thin Cylindrical Shell Share the Same Moment of Inertia as a Hoop?

[raycao88124]

Hi all
i am really confused about this, why does a thin cylindrical shell has the same moment of inertia of a hoop?
i understand the I for a thin hoop is mr(square), and i know how to do this. but i just get confused why a cylindrical shell has the same result? and i don't know how to show the work.
i asked my physics teacher but he didnt explain very clearly.

Moderation note: Duplicate threads merged since both threads had been answered.

 

[Hootenanny]

Hi all
i am really confused about this, why does a thin cylindrical shell has the same moment of inertia of a hoop?
i understand the I for a thin hoop is mr(square), and i know how to do this. but i just get confused why a cylindrical shell has the same result? and i don't know how to show the work.
i asked my physics teacher but he didnt explain very clearly.

Consider a thin hoop and a cylindrical shell with an axis of rotation through the centre of both circular faces (i.e. looking down the tube). In both cases, where is all the mass located? Can you see any difference between the hoop and the cylinder if viewed end-on?

 

[Redbelly98]

In both cases, the object's entire mass is concentrated at the same distance (r) from the axis of rotation. So I=mr²

What formula are you given to calculate I?

 

[rcgldr]

Why does a thin cylindrical shell has the same moment of inertia of a hoop?

Because the cross section of a cylinder is the same as a hoop. You could combine a bunch of hoops together to create a cylinder. The distribution of mass versus radius stays the same, only the amount of mass varies (the "m" in m r^2).