(a) Determine the acceleration of the center of mass of a uniform solid disk rol

Question

(a) Determine the acceleration of the center of mass of a uniform solid disk rolling down an incline making angle with the horizontal. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)
adisk =  

(b) Compare the acceleration found in part (a) with that of a uniform hoop. (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)
ahoop =  

(c) What is the minimum coefficient of friction required to maintain pure rolling motion for the disk? (Use any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)
=  

Explanation / Answer

A) net torque on disc Tnet = I*alpha = I*(a/R)

Tweight = m*g*R*sin(theta) = I*(a/R)

using parallel axis theorem

I = Icm + m*R^2 = (1/2)*mR^2 + m*R^2 = (3/2)*m*R^2

then m*g*R*sin(theta) = (3/2)*m*R^2*(a/R)

a = (2/3)*g*sin(theta)

B) similarly for hoop

Tnet = I*alpha = I*(a/R)

Tweight = m*g*R*sin(theta) = I*(a/R)

using parallel axis theorem

I = Icm + m*R^2 = mR^2 + m*R^2 = 2*m*R^2

then m*g*R*sin(theta) = 2*m*R^2*(a/R)

a = (1/2)*g*sin(theta)

C) The only force needed to provide tangential acceleration to a rolling body is the acceleration times the "effective" mass at the rim meff; for a disk meff = m/2.
So F(tangential) = meff*a = m/2 * 2/3*gsin(theta) = mgsin(theta)/3
Fnormal = mgcos(theta)
Ffric must be (at least) = Ftangential ==>
mu = Ffric/Fnormal = tan(theta)/3

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