The mechanism shown in the figure (Figure 1) is used to raise a crate of supplie

Question

The mechanism shown in the figure (Figure 1) is used to raise a crate of supplies from a ship's hold. The crate has total mass 45 kg. A rope is wrapped around a wooden cylinder that turns on a metal axle. The cylinder has radius 0.25 m and a moment of inertia I = 3.0 kg middot m^2 about the axle. The crate is suspended from the free end of the rope. One end of the axle pivots on frictionless bearings: a crank handle is attached to the other end. When the crank is turned, the end of the handle rotates about the axle in a vertical circle of radius 0.12 m, the cylinder turns, and the crate is raised. What magnitude of the force F applied tangentially to the rotating crank is required to raise the crate with an acceleration of 1.40 m/s^2? (You can ignore the mass of the rope as well as the moments of inertia of the axle and the crank.) Express your answer using two significant figures.

Explanation / Answer

Applying newton's law on crate

Fnet = m*a

in y-direction

T - mg = m*a

T = m*(g + a)

T = 45*(9.81 + 1.40) = 504.45 N

Force diagram for cyliner

torque = I*alpha

F*L - T*R = I*alpha

(alpha = a*R)

F*L = T*R + I*a/R

F = T*R/L + I*a/(R*L)

Using given values

F = 504.45*0.25/0.12 + 3*1.4/(0.25*0.12) = 1190.94 N

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