Rotational Sphere on Slope

• einstein18
In summary, the problem presented involves a solid sphere of mass M and radius R on an inclined plane with an angle of θ. The coefficient of static friction is μs and the sphere is assumed to roll without slipping down the plane. The correct equation for the acceleration of the center of mass of the sphere is a = gsinθ - μgcosθ, where gsinθ represents the acceleration due to gravity and μgcosθ represents the maximum static frictional force. The (2/5) factor is not included in the final equation as it is not relevant to the acceleration of the center of mass. The confusion was caused by not realizing that the question asked for the equation for the center of mass acceleration, not the

einstein18

Ok, I am trying to understand this problem on my practice exam and I can't figure out what I am doing wrong.

Homework Statement

A solid sphere of mass M and radius R is released from rest on an inclined plane with an angle of θ. The coefficient of static friction for the sphere on the plane is μs. Assuming that the sphere rolls without slipping down the plane and that the static frictional force is at its maximum value, which of the following is the correct equation for the acceleration of the center of mass of the sphere?

Homework Equations

Since it is static friction and the sphere doesn't slip:
X-axis: Mgsinθ = Fstatic
Y-axis: Mgcosθ = N
Torque: Fstatic*R = I*alpha
alpha = a/R
I of solid sphere = (2/5)MR^2

The Attempt at a Solution

Simplifying the torque equation and making substituitions:
a = Fstatic*R^2 / I -> a = (μMgcosθ)R^2 / (2/5)MR^2 ->

a = μgcosθ/(2/5)

However, the answer is: a = gsinθ - μgcosθ

I see where the gsinθ comes from but I don't understand why its there if μ is static.
Also the thing that is confusing me the most, where does (2/5) go?

Any help would be greatly appreciated,

I just realized that if I solve for the X-axis equation i get the answer:

mgsinθ - f = ma
a = gsinθ - μgcosθ

But this is only true if the friction were kinetic. Also where does the rotational aceleration go?

Oh! The question asks for the equation of the acceleration of the center of mass of the sphere. Doh. I can't believe i just spent the past hour trying to figure this out...

Einstein eh :P