An 878 kg car is held in place by an unreliable winch. The gearbox of the winch

Question

An 878 kg car is held in place by an unreliable winch. The gearbox of the winch breaks and at that moment the car drum system free falls from rest, shown in the figure. During the car's fall, there is no slipping betweern the (massless) rope, the pulley, and the winch drum. 26 kg.m2 39 cm 139 kg·m" 878 kg 9 m Find the speed of the car as it hits the ground. The moment of inertia of the winch drum is 139 kg. m2 and that of the pulley is 26 kg m2. The radius of the drum is 39 em and the radius of the pulley 6 cm and the acceleration due to gravity is 9.81 m/s Answer in units of m/s.

Explanation / Answer

Apply conservation of energy

As the car falls down its potential enrgy decreases and kinetic enrgy of car,winch drum and pulley increses.

loss in PE = gain in KE

m_car*g*h = 0.5*m_car*v^2 + 0.5*I1*w1^2(winch drum) + 0.5*I2*w2^2(pulley)

m_car*g*h = 0.5*m_car*v^2 + 0.5*I1*(v/r1)^2 + 0.5*I2*(v/r2)^2

m_car*g*h = 0.5*m_car*v^2 + 0.5*I1*v^2/r1^2 + 0.5*I2*v^2/r2^2

m_car*g*h = v^2*(0.5*m_car + 0.5*I1/r1^2 + 0.5*I2/r2^2)

v^2 = m_car*g*h/(0.5*m_car + 0.5*I1/r1^2 + 0.5*I2/r2^2)

= 878*9.81*9/(0.5*878 + 0.5*139/0.39^2 + 0.5*26/0.06^2)

= 17.2

v = sqrt(17.2)

= 4.15 m/s <<<<<<--------------Answer

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