1. A thin disk of mass 0.5 kg and radius 25 mm rolls on a surface with a coefficient of friction of 0.22. The center of the disk is being pulled by a constant force .

(a) What is the maximum value of such that the disk rolls without slip? Call this value .(b)

If the actual value of is 0.9 what is the acceleration of the mass center and the angular acceleration of the disk?(c) If the actual value of is what is the acceleration of the mass center and the angular acceleration of the disk? (d) For the situation described in part (c) hat is the kinetic energy of the disk 2 s after the force is applied?

a) FNmgGP

r = 25mm = 0.22m=0.5kgg=9.8m/s2 Thin Disk

If we assume that the disk rolls without slip then:

We can then determine the sum of the forces and the sum of the moments about G:

Based on the above free body diagram, these are then determined to be:

Simplify the sum of the moments and we get:

Plug F back into the equation for the sum of the forces:

Since we are assuming the disk rolls without slip:

Combine like terms and we get:

Rearrange and solve for :

Since we know what F is we can say:

Replacing we then have:

Simplify and solve for P:

Plugging in our known values we can then solve for Pcr:

b) We are told that P is now 0.9Pcr. This can be shown as:

Since P is less than Pcr, we know that the disk rolls without slip; so:

First, solve for :

We can then solve for :

c) We are told P is now 1.1Pcr. This can be shown as:

Since P is more than Pcr, we know that the disk rolls and slips; so:

There is no kinematic relationship here but we can use the sum of the forces from part a:

Simplify and solve for :

Using the sum of the moments we can solve for :

d) For kinetic energy we know:

We can integrate and to find and in terms of t. We can then solve at t=2sec so that we have:

We can find the moment of inertia with:

Plug these numbers into our original equation and we get: