A cable lifts a 1060-kg elevator at a constant velocity for a distance of 39 m. What is the work done by each of the following?
To determine the work done by different factors in this scenario, we need to understand the concept of work and how it is calculated.
Work (W) is defined as the product of force (F) applied on an object and the displacement (d) of that object in the direction of the force. Mathematically, it can be expressed as:
W = F * d * cos(theta)
Where:
- F is the force applied on the object
- d is the displacement of the object
- cos(theta) is the angle between the force and displacement vectors (if the force is applied at an angle)
Now, let's calculate the work done by each factor in this scenario.
1. The work done by the cable:
Since the elevator is being lifted at a constant velocity, it means there is no net acceleration. Therefore, the net force applied by the cable must be equal and opposite to the force of gravity acting on the elevator (to counterbalance it).
The force of gravity can be calculated using the formula:
F_gravity = m * g
Where:
- m is the mass of the elevator (1060 kg)
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
So, F_gravity = 1060 kg * 9.8 m/s^2 = 10388 N
As the elevator is being lifted vertically, the angle between the force and the displacement vectors is 0 degrees, meaning cos(0) = 1.
Therefore, the work done by the cable can be calculated as:
W_cable = F_cable * d * cos(theta)
= F_gravity * d * cos(0)
= 10388 N * 39 m * 1
= 404532 J (joules)
Hence, the work done by the cable is 404532 joules.
2. The work done by the net force:
As mentioned earlier, if the elevator is lifted at a constant velocity, the net force must be zero since there is no acceleration.
Therefore, the work done by the net force is also zero.
3. The work done by gravity:
Gravity is doing negative work on the elevator because the displacement is in the opposite direction of the gravitational force.
The work done by gravity can be calculated as:
W_gravity = -F_gravity * d * cos(theta)
= -10388 N * 39 m * cos(180 degrees)
= -404532 J (joules)
Since cos(180 degrees) = -1, the work done by gravity is -404532 joules.
To summarize:
- The work done by the cable is 404532 joules.
- The work done by the net force is zero.
- The work done by gravity is -404532 joules.