28- Consider the system shown in the figure below with m 1 = 23.0 kg, m 2 = 11.2

Question

28-Consider the system shown in the figure below with m1 = 23.0 kg, m2 = 11.2 kg, R = 0.100 m, and the mass of the pulley M = 5.00 kg. Object m2 is resting on the floor, and object m1 is 4.80 m above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley.

28-Consider the system shown in the figure below with m1 = 23.0 kg, m2 = 11.2 kg, R = 0.100 m, and the mass of the pulley M = 5.00 kg. Object m2 is resting on the floor, and object m1 is 4.80 m above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley.

Explanation / Answer

Use the energy equation
Initial Energy = Final Energy

Initial Energy is only the potential energy of mass one (PE1).
Final Energy is the final kinetic energy of mass one (KE1), the final kinetic energy of mass two (KE2), the kinetic energy of the pulley (KEp) and the potential energy of mass two (PE2)

PE1 = KE1 + KE2 + KEp + PE2

PE1 = m1*g*h
PE2 = m2*g*h
KE1 = (1/2) m1 * v^2
KE2 = (1/2) m2 * v^2
KEp = (1/2) I w^2
I = (1/2) M r^2

and we know that w = (v/r)
PE1 = KE1 + KE2 + KEp + PE2
m1*g*h = (1/2) m1 * v^2 + (1/2) m2 * v^2 + (1/2) I w^2 + m2*g*h
Now substitute in I = (1/2) M r^2 and w = (v/r)
m1*g*h = (1/2) m1 * v^2 + (1/2) m2 * v^2 + (1/2) (1/2) M r^2 (v/r)^2 + m2*g*h
m1*g*h = (1/2) m1 * v^2 + (1/2) m2 * v^2 + (1/4) M v^2 + m2*g*h
m1*g*h - m2*g*h = v^2 [(1/2) m1 + (1/2) m2 + (1/4) M]
v^2 = (m1*g*h - m2*g*h)/[(1/2) m1 + (1/2) m2 + (1/4) M]
v^2 = g*h*(m1 - m2)/[(1/2) m1 + (1/2) m2 + (1/4) M]
v = sqrt[g*h*(m1 - m2)/[(1/2) m1 + (1/2) m2 + (1/4) M]]
v = sqrt[9.81 m/s^2 *4.80 m *(23.00 kg - 11.20 kg)/[(1/2) 23.00 kg + (1/2) 11.2 kg + (1/4) 5.00 kg]]
v = 5.5 m/s
There is one motion equation without acceleration
h = h0 + (1/2) * (v - v0) * t
h0 = 0 m
v0 = 0 m/s
h = v*t/2
t = 2 * h / v
t = 2 * 4.80 m / 5.5 m/s
t = 1.7445 sec

(b) This is the same up to here
PE1 = KE1 + KE2 + KEp + PE2
There is no KEp, everything else is the same.
So v = sqrt[g*h*(m1 - m2)/[(1/2) m1 + (1/2) m2]]
v = sqrt[9.81 m/s^2 *4.80 m *(23.00 kg - 11.20 kg)/[(1/2) 23.00 kg + (1/2) 11.2 kg]]
v = 5.7 m/s
t = 2 * h / v
t = 2 * 4.80 m / 5.7 m/s
t = 1.684 sec

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