A construction crew pulls up an 87.5-kg load using a rope thrown over a pulley and pulled by an electric motor. They lift the load 15.1 m and it arrives with a speed of 15.6 m/s having started from rest. Assume that acceleration was not constant.
a. How much work (J) was done by the motor?
b. How much work was done by gravity?
c. What constant force N could the motor have exerted to cause this motion?
To answer these questions, we need to apply the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. In this case, we can break down the problem into two parts: the work done by the motor and the work done against gravity.
a. To find the work done by the motor, we need to calculate the change in kinetic energy. The initial kinetic energy is zero because the load starts from rest. The final kinetic energy can be calculated using the equation:
Kinetic energy = (1/2) * mass * velocity^2
Given that the mass is 87.5 kg and the final velocity is 15.6 m/s, we can calculate the final kinetic energy.
Final Kinetic energy = (1/2) * 87.5 kg * (15.6 m/s)^2
b. To find the work done against gravity, we need to calculate the gravitational potential energy. The gravitational potential energy is given by:
Gravitational potential energy = mass * gravitational acceleration * height
Given that the mass is 87.5 kg, the gravitational acceleration is approximately 9.8 m/s^2, and the height is 15.1 m, we can calculate the gravitational potential energy.
Gravitational potential energy = 87.5 kg * 9.8 m/s^2 * 15.1 m
c. The constant force exerted by the motor is equivalent to the net force acting on the load. The net force can be calculated by using Newton's second law, which states that the net force is equal to the mass times the acceleration.
Net force = mass * acceleration
Since the acceleration is not constant, we need to use an average acceleration. We can calculate the average acceleration by using the equation:
Average acceleration = (final velocity - initial velocity) / time
However, we are not given the time it took for the load to reach its final velocity, so we cannot directly calculate the average acceleration. Therefore, we cannot determine the constant force exerted by the motor without additional information.
To summarize:
a. The work done by the motor can be calculated using the final kinetic energy.
b. The work done against gravity can be calculated using the gravitational potential energy.
c. The constant force exerted by the motor cannot be determined without additional information.