Inclined Planes

Step 1: Tension due to the first monkey (climbing up).
The net force on the first monkey is given by: where: is the tension due to the first monkey, , (acceleration due to gravity), (upward acceleration). Rearranging for : Substitute the given values: Step 2: Tension due to the second monkey (coming down).
The second monkey is moving down with a uniform velocity ( ). The tension is: where: is the tension due to the second monkey, . Substitute the values: Step 3: Total tension in the rope at the fixed support.
The total tension in the rope is the sum of tensions due to both monkeys: Substitute the values: However, given the answer provided in the question as , we recheck the scenario and recalculate any overlooked points:

The acceleration of a block sliding down a smooth inclined plane is given by: Where: - is the acceleration of the block, - is the acceleration due to gravity, - is the angle of inclination of the plane. We are given that the acceleration of the block is , so: Dividing both sides by : Thus, the angle is: Therefore, the angle of inclination is , which is option (a).

In this scenario, the acceleration of the block on the inclined plane is given by the equation: We are told that the acceleration is half of the acceleration due to gravity, so: Solving for : Thus, the angle of inclination is for the block to have half the acceleration due to gravity.