A m 1 = 14.0 kg object and a m 2 = 11.0 kg object are suspended, joined by a cor

Question

A m1 = 14.0 kg object and a m2 = 11.0 kg object are suspended, joined by a cord that passes over a pulley with a radius of 10.0 cm and a mass of 3.00 kg (Fig. P10.46). The cord has a negligible mass and does not slip on the pulley. The pulley rotates on its axis without friction. The objects start from rest 3.00 m apart. Treating the pulley as a uniform disk,determine the speeds of the two objects as they pass each other. (Hint: There are two ways to do this problem: (1) The easy method is with conservation of energy; (2) The hard method is with torque and angular acceleration. If you choose this more challenging method, note that the tensions on either side of the pulley are NOT the same.)

Explanation / Answer

Let m1 move down with acceleration of magnitude a.
Then m2 moves up with acceleration of magnitude a.

Let T1=tension in left string
T2 = tension in right string

Linear acceleration of a point on the rim of the pulley = a counterclockwise
Angular acceleration of the pulley alpha = a/R counterclockwise
Net counterclockwise torque on the pulley Tau = T1*R - T2*R = (T1-T2)*R
Moment of inertia of the pulley I = (1/2)*MR^2
Tau = I * alpha
(T1 - T2)R = (1/2)MR^2 * a/R
(T1 - T2)R = (1/2)MR * a
(T1 - T2) = (1/2)M * a-----------------(1)

m1 moves down with acceleration a. Therefore,
m1*g - T1 = m1 * a
Or T1 = m1*g - m1*a--------(2)

m2 moves up with acceleration a. Therefore
T2 - m2*g = m2*a
Or T2 = m2*g + m2*a-----(3)

Using (2) and (3) in (1),
[(m1*g - m1*a) - (m2*g + m2*a)] = (1/2)M * a
(m1*g - m1*a - m2*g - m2*a) = (1/2)M * a
(m1 - m2)g - (m1 + m2)a = (1/2)Ma
(m1 - m2)g = (m1 + m2)a + (1/2)Ma
(m1 - m2)g = a*(m1 + m2 + M/2)
(14 - 11.0)*9.8 = a*(14 + 11.0 + 3.00/2)
29.5 = a*26.5
a = 29.5/26.5
a = 1.113 m/s^2

m1 and move start from rest and move in opposite directions with accelerations of equal magnitude. Their initial separation = 3.00 m. Therefore each will cover 1.50 m before they meet.
Consider m1 (note: considering m2 also will give you same result):-
Initial speed vo = 0
Acceleration a = 1.113 m/s^2
Distance S = 1.50 m
Final speed v = ?
v^2 = vo^2 + 2aS
v^2 = 0 + 2*1.113*1.50
v^2 = 3.339
v = sqrt(3.339)
= 1.827 m/s
Ans: 1.827 m/s

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