Help! Moment of Inertia for Tangent Axis of Sphere

[envscigrl]

Help! tangent axis

Use the parallel-axis theorem to find the moment of inertia of a solid sphere of mass M=3.80kg and radius R=2.30m about an axis that is tangent to the sphere.
I am being thrown off this problem because the axis is TANGENT to the sphere and not through the center. According to the equations in my book i thought the equation I would use would be:
I= Icm +Mr^2= 3/2 Mr^2
but it didnt work! am i not using the parallel axis right??
Thanks for helping!

 

[maverick280857]

If memory serves me, the moment of inertia of a solid sphere about an axis passing through the center of mass is

I_{cm} = \frac{2}{5}MR^2

Add to it MR^2 to get the answer. How are you getting \frac{3}{2}MR^2??

Remember an axis parallel to one passing through the sphere is clearly a tangent to the sphere from geometry. The parallel axis transformation changes your axes all right but the new axis of rotation is parallel to the one you started out with. So that shouldn't be a problem.

Cheers
Vivek

 

[wais]

Don't worry, it's understandable to get confused when dealing with different types of axes. In this case, since the axis is tangent to the sphere, it is not parallel to the axis through the center of mass. Therefore, you cannot use the parallel-axis theorem as you mentioned. Instead, you will need to use the perpendicular-axis theorem to find the moment of inertia about the tangent axis. This theorem states that the moment of inertia about an axis perpendicular to the plane of motion is equal to the sum of the moments of inertia about two perpendicular axes through the same point. In this case, the two perpendicular axes would be the axis through the center of mass and the tangent axis. So, the equation you would use is:

I = Icm + Iparallel

Where Icm is the moment of inertia about the axis through the center of mass, and Iparallel is the moment of inertia about the tangent axis. Since the moment of inertia about the axis through the center of mass for a solid sphere is 2/5 MR^2, the equation would be:

I = (2/5 MR^2) + Iparallel

Now, to find the moment of inertia about the tangent axis, you can use the perpendicular-axis theorem again, but this time with the axis through the center of mass and an axis perpendicular to the tangent axis. This perpendicular axis would be the axis passing through the center of the sphere and the point of tangency of the tangent axis. The moment of inertia about this axis is simply MR^2. So, the equation becomes:

I = (2/5 MR^2) + (MR^2)

= (7/5 MR^2)

= (7/5)(3.80kg)(2.30m)^2

= 46.15 kg*m^2

Therefore, the moment of inertia about the tangent axis for this solid sphere is 46.15 kg*m^2. I hope this helps clear up any confusion and helps you solve the problem!