Question

Problem 1. (10 pts) Consider a pulley that is well-approximated as a vertical disc of radius R = 0.125 m and mass M = 4.1 kg that can rotate about an axis through its center. A massless, inextensible string is hung over the pulley and two small masses mı = 0.15 kg and m2 = 0.20 kg are hung, one on each end of the string. They are released and the pulley begins to rotate as the small masses accelerate, one up and the other down. bo a) Draw the picture and draw free-body diagrams for the three objects. b) Use forces and torques to determine the angular acceleration of the disc and the linear accel- erations for my and m2.

Answer 1

Let the linear accelration of both the masses be "a"

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And angular acceleration of pulley be "x"

For pulley I = mr²/2 (disk),

a = (x)(r) (pure rolling motion)

From FBD of m1 ,

T1 -- m1g = m1a ......(1)

From FBD of m2,

m2g - T2 = m2a ........(2)

From FBD of pulley,

T2r -T1r = I(x) = (mr²/2)(x) = mra/2

T2 - T1 = ma/2 ......(3)

Adding equation 1 &2&3 we get

(m2 -m1)g = a(m1+m2 +M/2)

a = (m2 - m1)g/ (m1 +m2+M/2)

x = a/r = (m2 -m1)g/[(m1+m2+M/2)(r)]

Putting value of m1 = 0.15kg , m2 = 0.2 kg , M 4.1 kg ,

r= 0.125m , g= 9.8m/s² ,

we get a =0.20m/s² x = 1.63rad/s²

Angular acceleration (x)= 1.63rad/s²

Linear acceleration (a) = 0.20m/s²