Rigid Body Motion: Sphere on Inclined Plane

[hellsnake]

Homework Statement

A sphere of mass M and radius R it`s given a velocity Vo on the base of a inclined plane (theta being the angle) and friction coefficient mu (you may assume that static and kinetic friction are equal) Find the position of the ball as a function of time

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Welcome to PF!

Hi hellsnake! Welcome to PF!

(have a theta: θ and a mu: µ )

(and btw, is the sphere skidding, or is it rolling without slipping?)

Use work done and conservation of energy …

what do you get? ​

 

[hellsnake]

It doesn'st tell you if it's slipping or not you have to see that. By the way I forgot you have to study the two cases when µ≤ 2/7 tan θ and µ≥ 2/7 tan θ. Without the equal. And when I do conservation of energy and work done by friction I get this differential equation
(1/2)M.(Vcm)² + (1/2)Icm.w² + MgH - (1/2)MVo² = -Ffr X(t)
But Vcm= (d/dt)X(t) and H=X(t)sin θ . Then I get that

(1/2)M.(d/dt)X(t))² + (1/2)Icm.w² + MgX(t)sin θ - (1/2)MVo² = -Ffr X(t)

Where:
M= mass of the sphere
Vcm= It's the velocity of the center of the mass
Icm= It's the moment or inertia with an axis that pass through the center oh the sphere
w= It's the angular velocity
H= It's how high is the sphere measure from the floor
Vo= It's the initial velocity
Ffr= It's the force that the frcition does
X(t)= It'sthe position of the ball measure from the point where it start his motion. And I choose the X axis parallel to the motion

So if I read the equation I get that the energy on a arbitrary point of the sphere is equal to
(1/2)M.(Vcm)² + (1/2)Icm.w² + MgH. Now this minus the energy in the initial moment(the sphere doesn't have rotational energy here) (1/2)MVo² . All of this is equal to the work done by friction -Ffr X(t)

I think that I have to do another thing because this differential equation It's too hard for this course I think. However if you have the solution for this equation I appreciate it too

Thanks a lot in advance

 

[tiny-tim]

Hi hellsnake!

(have an omega: ω )

You can simplify it slightly by writing I_cm in terms of M and R, and writing F_fr = µMg.

After that, you need a relation between V and ω … try the easier, rolling-without-slipping case first: V = Rω.

(and after that, you'll need to find the condition on µ for rolling … presumably it's going to be µ ≥ 2/7 tanθ ! )