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?
increase in potential energy = A = m g h
increase in kinetic energy = B = (1/2) m v^2
total work done by motor = A+B
work done by gravity = -A
F = (A+B)/15.1
By the way, do not forget g (9.81 m/s^2) in m g h
To find the answers to these questions, we can use the work-energy principle and the concept of mechanical work.
a. The work done by the motor can be calculated using the work-energy principle:
Work done by the motor = Change in kinetic energy
= (1/2) * mass * (final velocity^2 - initial velocity^2)
= (1/2) * 87.5 kg * (15.6 m/s)^2
b. The work done by gravity can be calculated considering the force of gravity acting on the load:
Work done by gravity = force of gravity * distance
= mass * gravitational acceleration * distance
= 87.5 kg * 9.8 m/s^2 * 15.1 m
c. The constant force that the motor could have exerted can be found by using the equation of motion:
Force = mass * acceleration
Since acceleration is not constant, we need to find the average force exerted by the motor.
Average force = Change in momentum / time
= mass * (final velocity - initial velocity) / time
= 87.5 kg * (15.6 m/s - 0 m/s) / time
Note: The time is not provided in the given information, so we cannot directly calculate the constant force.
Please provide the time taken to lift the load, and I will be able to help you with the calculation of the constant force.
To solve this problem, we need to analyze the forces involved and apply the laws of motion. Let's break down each part of the problem and find the solutions step by step.
a. How much work (J) was done by the motor?
Work is defined as the product of force and displacement in the direction of the force. In this case, the motor exerts a force to lift the load. The work done by the motor is equal to the change in kinetic energy of the load.
The initial kinetic energy (KEi) is zero because the load starts from rest. The final kinetic energy (KEf) is given by:
KEf = (1/2)mv^2
where m represents the mass (87.5 kg) of the load, and v represents the final velocity (15.6 m/s) of the load.
Therefore, the work done by the motor is given by:
Work = KEf - KEi = (1/2)mv^2 - 0 = (1/2)(87.5 kg)(15.6 m/s)^2
You can calculate the numerical value of this expression to find the work done by the motor in joules (J).
b. How much work was done by gravity?
Gravity acts on the load and does work as the load is lifted vertically. The work done by gravity is equal to the gravitational potential energy gained by the load.
The gravitational potential energy (PE) is given by:
PE = mgh
where m represents the mass (87.5 kg) of the load, g represents the acceleration due to gravity (9.8 m/s^2), and h represents the vertical displacement (15.1 m) of the load.
Therefore, the work done by gravity is given by:
Work = mgh = (87.5 kg)(9.8 m/s^2)(15.1 m)
You can calculate the numerical value of this expression to find the work done by gravity in joules (J).
c. What constant force (N) could the motor have exerted to cause this motion?
The force exerted by the motor is not constant, but we can find its average force by dividing the work done by the motor (from part a) by the distance traveled.
Average Force = Work / Distance
Average Force = (1/2)(87.5 kg)(15.6 m/s)^2 / 15.1 m
You can calculate the numerical value of this expression to find the average force exerted by the motor in newtons (N).
Please note that the above calculations assume that there are no other forces (such as friction) acting on the load.