A monkey of mass m climbs a rope slung over a light frictionless pulley.The opposite end of the rope is tied to a block of Mass M lying on a smooth horizontal plane.Find the acceleration of the block and tension in the rope in the following cases
1)the monkey moves up with an acceleration 'a" relative to the rope.
2)the monkey moves downward with an acceleration 'b" relative to the rope
To find the acceleration of the block and the tension in the rope in these two cases, we can use Newton's second law and the concepts of relative motion.
1) When the monkey moves up with an acceleration 'a' relative to the rope:
In this case, the monkey's acceleration will also be the same as the acceleration of the block since they are attached by the rope.
Let's denote the upward acceleration of the monkey and the block as 'A'. To find 'A', we can apply Newton's second law separately to the monkey and the block.
For the monkey:
The net force acting on the monkey is the tension in the rope (T) minus the weight of the monkey (mg), where 'g' is the acceleration due to gravity:
T - mg = Ma
For the block:
The net force acting on the block is the tension in the rope (T) minus the weight of the block (Mg):
T - Mg = Ma
Since the monkey and the block have the same acceleration 'A', we can write:
Ma = Ma + Mg
Simplifying the equation, we get:
Mg = 0
This means that the block will not accelerate, and its acceleration 'A' will be zero. Therefore, the tension in the rope will be equal to the weight of the block: T = Mg.
2) When the monkey moves downward with an acceleration 'b' relative to the rope:
Similarly, we can apply Newton's second law to find the acceleration of the block and the tension in the rope.
For the monkey:
The net force acting on the monkey is the weight of the monkey (mg) minus the tension in the rope (T):
mg - T = Ma
For the block:
The net force acting on the block is the weight of the block (Mg) minus the tension in the rope (T):
Mg - T = Ma
Since the monkey and the block have the same acceleration 'A', we can write:
mg - T = Ma
Mg - T = Ma
Simplifying the equations, we get:
T = mg - Ma
T = Mg - Ma
In this case, both the monkey and the block will have an acceleration 'A', and the tension in the rope will be determined by the difference between the weight of the monkey (mg) and the weight of the block (Mg).
These are the calculations to find the acceleration of the block and the tension in the rope in the two given cases.