- #1

songoku

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## Homework Statement

A disk of radius r and mass m is placed on an incline which makes an angle of 30 degree. The disk is suspended by a string attached to its center and a mass 3m is hung at the other end. The surface of incline is rough with the coefficient of static and kinetic friction denoted as μ

_{s}and μ

_{k}respectively. Assume that the string and the pulley does not have any friction and the pulley does not rotate.

a. find the linear acceleration of the disk when the disk rolls without slipping

b. for (a) to happen, find the condition of μ

_{s}

c. when the condition of (b) is not met, the disk starts rolling but with sliding. Find a relative acceleration between the linear and angular acceleration, i.e. sliding acceleration against the surface of incline

## Homework Equations

torque = I . α

∑F = m.a

## The Attempt at a Solution

a.

For the 3m object:

∑F = m.a

W - T = 3ma

3mg - T = 3ma ... (1)

For the disk:

∑F = m.a

T - f - W sin θ = m.a

T - μ

_{k}N - mg sin θ = m.a

T - μ

_{k}. mg cos θ - mg sin θ = m.a ... (2)

I just need to solve for a from equation (1) and (2). Is this correct?b.

Στ = I.α

f . r = 1/2 mr

^{2}. a/r (I assume moment of inertia of disk is 1/2 Mr

^{2}

μ

_{s}. mg cos θ . r = 1/2 mr

^{2}. a/r (I am not sure about this part. The disk is moving so why don't we use μ

_{k}instead?)

μ

_{s}= 0.5 a / (g cos θ) then I substitute the acceleration I got from part (a). Is this correct?

c. I am not sure what the question asks me to do. Won't the equation be the same as part (a)? What is "relative acceleration between the linear and angular acceleration"? Is it linear acceleration minus angular acceleration?

Thanks