Question:

A 1 kg mass is lifted by a rope and pulled vertically upwards with a constant speed of 2 m/s. The rope is attached to a pulley of radius 0.5 m, which has a moment of inertia of 0.2 kg·m² and is initially at rest. Neglecting any friction or air resistance, determine the power exerted by the person pulling the rope.

(a) What is the net force acting on the mass while it is being lifted?

(b) What is the tension in the rope while the mass is being lifted?

(c) What is the angular acceleration of the pulley while the mass is being lifted?

(d) How much work is done by the person pulling the rope in lifting the mass by 3 meters?

(e) Determine the power exerted by the person pulling the rope.

Answer:

(a) The net force acting on the mass while it is being lifted can be determined using Newton's second law, which states that the net force is equal to the product of mass and acceleration. Since the speed is constant, the acceleration is zero, and therefore, the net force is also zero.

Net force = 0 N

(b) The tension in the rope can be determined using the equation:

Tension = force = mass * acceleration = mass * g

where g is the acceleration due to gravity (9.8 m/s²).

Tension = 1 kg * 9.8 m/s² = 9.8 N

(c) The angular acceleration of the pulley can be determined using the equation:

Torque = moment of inertia * angular acceleration

Since the pulley is initially at rest and the rope is pulled vertically upwards, the torque is equal to the tension in the rope multiplied by the radius of the pulley.

Torque = Tension * radius

Using the value of tension calculated in part (b) and the given radius, the torque becomes:

Torque = 9.8 N * 0.5 m = 4.9 N·m

Now, we can solve for angular acceleration:

4.9 N·m = 0.2 kg·m² * angular acceleration

angular acceleration = 4.9 N·m / 0.2 kg·m² = 24.5 rad/s²

(d) The work done by the person pulling the rope can be determined using the equation:

Work = force * distance

Since the force required to lift the mass is equal to its weight (mass * g), the work done becomes:

Work = mass * g * distance

Work = 1 kg * 9.8 m/s² * 3 m = 29.4 J

Therefore, the person has done 29.4 Joules of work in lifting the mass by 3 meters.

(e) Power can be determined using the equation:

Power = work / time

Since the speed is constant and the distance is given, we can calculate the time using the equation:

time = distance / speed

time = 3 m / 2 m/s = 1.5 s

Now we can calculate the power:

Power = 29.4 J / 1.5 s = 19.6 W

Therefore, the power exerted by the person pulling the rope is 19.6 Watts.