A chandelier of mass 5M is being installed in the new opera house at a height h

Question

A chandelier of mass 5M is being installed in the new opera house at a height h above the floor. The chandelier is connected to one end of a steel cable with negligible mass which runs over a pulley of mass M, radius R, and moment of inertia 1/2 MR2. The other end of the cable is pulled by the installation crew with constant force F (see figure). Write your results in terms of M, h, R, and F. Remember to check the dimensions/units for each answer. Find the angular acceleration of the pulley. How long will it take for the pulley to make the first full loop (i.e., starting from rest)? What is the force exerted by the cable on the chandelier?

Explanation / Answer

Here .

a) let the angular acceleration of the pulley is a

using second law of motion

acceleration of the chandelier = (F - 5 M * g)/(5 M + 0.5 * M * r^2/r^2)

acceleration = (F - 5 M * g)/(5 M + 0.5 * M)

acceleration = (F - 5M*g)/(6.5 * M)

a = acceleration/R

a = (F - 5M*g)/(6.5 * M*R)

the angular acceleration is (F - 5M*g)/(6.5 * M*R)

b)

time talen is t

using second equation of motion

2pi = 0.5 * a * t^2

2*pi = 0.5 * (F - 5M*g)/(6.5 * M*R) * t^2

t = sqrt(4 *pi * (6.5 * M*R)/(F - 5*M*g))

the time taken is sqrt(4 *pi * (6.5 * M*R)/(F - 5*M*g))

c)

let the force on the chandelier is F1

F1 - 5 * M* g = 5 (F - 5M*g)/(6.5 *R)

F1 = 5M*g + 5 (F - 5M*g)/(6.5 *R)

the force exerted by the cable is 5M*g + 5 (F - 5M*g)/(6.5 *R)

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