Sphere rolling down an incline

[dk214]

Homework Statement

A hollow spherical shell with mass 2.50 kg rolls without slipping down a slope that makes an angle of 32.0 degrees with the horizontal.
a. Find the magnitude of the acceleration $$a_c$$ of the center of mass of the spherical shell.
b. Find the magnitude of the frictional force acting on the spherical shell.
c. Find the minimum coefficient of friction $$\mu$$ needed to prevent the spherical shell from slipping as it rolls down the slope.

Homework Equations

For part a.
Since its pure roll, $$a_c = \alpha * R \alpha = a_c/R$$
$$\tau = R*Friction = I (moment-of-inertia) * \alpha$$
$$Friction = (I*\alpha)/R = (I*a_c)/R^2$$
$$Ma_c = Mgsin(\theta)-Friction$$
$$Ma_c = Mgsin(\theta)-Ia_c/R^2$$
$$a_c = (MR^2*g*sin(\theta))/(MR^2+I)$$

The Attempt at a Solution

I for sphere =$$2/3 MR^2$$
so, $$a_c = (MR^2*g*sin(\theta))/(MR^2+2/3*MR^2)$$
MR^2 cancels..
$$a_c = 3/5*g*sin(\theta)$$
for a_c i got $$a_c = 3.12m/s^2$$ i think I am right unless i made a mathematical error some where.
and substituting a_c, in $$Ma_c = Mgsin(\theta)-Friction$$
i got Friction = 5.19 N.
And c,
this where I'm kind of stuck. I'm assuming since they are asking for minimum $$\mu$$ Friction is 0 in [tex] Ma_c = Mgsin(\theta)-Friction [\tex]
$$a_c = gsin(\theta)$$.
$$Friction = (I*\alpha)/R = (I*a_c)/R^2$$, and
$$Friction = \mu*mg*sin(\theta)$$
[tex] /mu= ((I*a_c)/R^2)/mg*sin(\theta)
idk if I'm right in assuming Friction is 0 in one part and not in other.. Any hints/guides and help would greatly be appreciated.

 

[alphysicist]

Hi dk214,

The Attempt at a Solution

I for sphere =$$2/3 MR^2$$
so, $$a_c = (MR^2*g*sin(\theta))/(MR^2+2/3*MR^2)$$
MR^2 cancels..
$$a_c = 3/5*g*sin(\theta)$$
for a_c i got $$a_c = 3.12m/s^2$$ i think I am right unless i made a mathematical error some where.
and substituting a_c, in $$Ma_c = Mgsin(\theta)-Friction$$
i got Friction = 5.19 N.
And c,
this where I'm kind of stuck. I'm assuming since they are asking for minimum $$\mu$$ Friction is 0 in

I don't think this is right; you've already found the force of friction. Now they want the minimum $\mu$ that can supply that force; in other words they want the coefficient for which that frictional force is a maximum. What does that give?

 

[dk214]

I don't know if I'm understanding the question right. Are they just asking for the $$\mu$$ for the friction I found.?
which would just be Friction/Normal
$$\mu = 5.12/Mgcos(\theta)$$
$$\mu = .246$$