**1. The problem statement, all variables and given/known data**
A solid sphere of mass 3.00 kg and radius 12.5cm rolls without slipping down an incline of

angle 13.5 degree for 250 m. Find the minimum coefficient of static friction required for a

rolling without slipping. What is the velocity of the center of the sphere at the bottom of the

incline? What is the angular momentum at that point? That is the kinetic energy at this point? Make a drawing, show the forces and torques. Indicate the torque which you are using for your calculations. Derive your formulas.

[Figures can all be rounded to 3 sig figs.]

M = 3.00 kg

R = 12.5 cm = 0.123 m

theta = 13.5

d = 250 m

h = d*sin(theta) = 58.4 m

**2. Relevant equations**
V = volume = 4/3(pi)R

^{3}; dV/dR = 4(pi)R

^{2}
(Rho) = M/V

I

_{cm} = (integral)r

^{2}dm

dm = (Rho)*dV

ME = K

_{rot} + K

_{cm} - U

L=I

_{a}(omega); I

_{a} = I

_{cm} + Md^2 where d is the distance between the two axes

**3. The attempt at a solution**
I found the velocity fine, and the answer matched with the solutions, but I can't seem to get the angular momentum, coefficient of friction, or the kinetic energy at the bottom.

__For friction...__
I know that f

_{s} = µ

_{s}*N & N = Mgcos(theta) but how do I find out what f

_{s} is?

__For angular momentum...__
I started with the L equation above, then subbed in the values from my previously derived work (see image) but the answer I got was way off from the correct answer.

__For kinetic energy at the bottom...__
It would just be 7/10*M*v

^{2} minus the work done by friction correct? So if I can somehow find the work done by friction over the distance traveled, then I find the amount of kinetic energy left at the bottom... would that be right?

I know there are a lot of questions here, but since they're all related to the same problem, I thought I would take a shot and just put all the thoughts in my head about this problem and just have the community pick at what they feel they want to attack first, and maybe help me make a game plan for this sort of problem. Thanks in advance for any help given!

Official Answers:

V=28.6 m/s

L=4.39 kg m

^{2}/s

µ

_{s}=0.0686