Two boxes hang from a solid disk pulley that is free to rotate as the blocks ris

Question

Two boxes hang from a solid disk pulley that is free to rotate as the blocks rise/fall. The left box has a mass m1 = 4.7 kg and the right box has a mass m2 = 1.9 kg. The pulley has mass m3 = 2.9 kg and radius R = 0.12 m.

1) What is the linear acceleration of the left box? (up is the positive direction)

2) What is the angular acceleration of the pulley? (let counter-clockwise be positive)

3) What is the tension in the string between the left mass and the pulley?

4) What is the tension in the string between the right mass and the pulley?

5) The boxes accelerate for a time t = 1.71 s. What distance does each box move in the time 1.71 s?

6) What is the magnitude of the velocity of the boxes after the time 1.71 s?

7) What is the magnitude of the final angular speed of the pulley?

Explanation / Answer

1) From newton's second law a(M + m) + I alpha/r = F

a = (M - m)g / [(M + m) + (1/2 M r2)/r2] = (M - m)g / [(M + m) + 1/2 Md ]

= [(4.7 - 1.9)9.8] / [(4.7 + 1.9) + 2.9/2 ] = 3.41 m/s2

2) alpha = a / r

= 3.41 / 0.12 = 28.4 rad/s2

3) T1 = M (g - a)

= 4.7(9.8 - 3.41) = 30.03 N

4) T2 = m (g + a)

= 1.9(9.8 + 3.41) = 25.1 N

5) s = 1/2 a t2

= 1/2 * 3.41 * 1.712= 4.98 m

6) v = sqrt [2 a s]

= sqrt [2 * 3.41 * 4.98] = 5.83 m/s

7) w = v / r

= 5.83 / 0.12 = 48.58 rad/s

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