Figure 5-53 shows a man sitting in a bosun's chair that dangles from a massless rope, which runs over a massless, frictionless pulley and back down to the man's hand. The combined mass of man and chair is 95.4 kg. With what force magnitude must the man pull on the rope if he is to rise (a) with a constant velocity and (b) with an upward acceleration of 1.46 m/s2? (Hint: A free-body diagram can really help.)
If the rope on the right extends to the ground and is pulled by a co-worker, with what force magnitude must the co-worker pull for the man to rise (c) with a constant velocity and (d) with an upward acceleration of 1.46 m/s2? What is the magnitude of the force on the ceiling from the pulley system in (e) part a (f) part b, (g) part c, and (h) part d?
(a)
2T-Mg=Ma
a=0
solve for T
T=mg/2
To solve this problem, we will break it down into several parts:
(a) Constant velocity:
When the man rises with a constant velocity, it means that the upward force he exerts on the rope is equal to the force of gravity pulling him downward. We can calculate this force using Newton's second law: F = mass * acceleration.
The force the man needs to pull to rise with a constant velocity is given by:
Force (a) = Mass * Acceleration
= 95.4 kg * 9.8 m/s^2 (acceleration due to gravity)
= 934.92 N
(b) Upward acceleration:
When the man rises with an upward acceleration of 1.46 m/s^2, we need to consider both the force to overcome gravity and the force required to accelerate him upward. In this case, the net force acting on the man will be the sum of the force needed to counteract gravity and the force required for acceleration.
The net force can be calculated as:
Net Force (b) = Mass * (Acceleration due to gravity + Upward acceleration)
= 95.4 kg * (9.8 m/s^2 + 1.46 m/s^2)
= 1155.444 N
(c) Constant velocity with a co-worker pulling:
When the rope is pulled by a co-worker, the force required by the man to rise with a constant velocity will be reduced. It will be equal to the difference between the force of gravity and the force exerted by the co-worker.
Force (c) = Force of gravity - Force exerted by the co-worker
= Mass * Acceleration due to gravity - Force exerted by the co-worker
(d) Upward acceleration with a co-worker pulling:
In this case, the net force acting on the man will be the sum of the force required for acceleration upward and the force exerted by the co-worker.
Net Force (d) = Mass * (Acceleration due to gravity + Upward acceleration) - Force exerted by the co-worker
(e), (f), (g), (h):
To find the magnitude of the force on the ceiling, we need to consider the forces acting on the entire system and apply Newton's third law (action-reaction pairs). The force exerted by the rope on the ceiling is equal in magnitude but opposite in direction to the force exerted by the ceiling on the rope.
Therefore, the magnitude of the force on the ceiling is equal to the force the man exerts on the rope when rising with a constant velocity or upward acceleration, as calculated in parts (a) to (d).