assume a frictionless ramp; however the pulley does have an unknown coefficient

Question

assume a frictionless ramp; however the pulley does have an unknown coefficient of static friction . Mass m=7.0 kg and mass=20.0 kg . The pulley has a mass of m=2.00 kg and a radius of R=0.05 m. The rope does not slip as it rotates around the pulley. Assume the moment of interia of the pulley is that of a solid cylinder of radius R .

1- determine the acceleration of the system .

2- what is the torque on the pulley ?

3-how much work is done by the torque as the pulley rotates through one revolution ?

Explanation / Answer

Writing force equation on mass M

M*g - T1 = Ma

Now, for mass m

T2 - mg = ma

So, adding both equations:

Mg - mg +(T2-T1) = (M+m)*a

For the pulley:

(T1-T2)*R = (0.5*Mp*R^2)*(a/R)

So, T1 - T2 = 0.5*Mp*a

So, (M-m)*g - 0.5*Mp*a = (M+m)*a

So, (M-m)*g = (M+m+Mp/2)*a

So, a = (M-m)*g/(M+m+Mp) = (20-7)*9.8/(20+7+2/2) = 4.55 m/s2

b)

Torque on pulley = (T1- T2)*R = 0.5*Mp*a*R = 0.5*2*4.55*0.05 = 0.2275 N.m

c)

Work done by torque = 0.2275*2*pi = 1.43 J

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