An electric elevator cable with a motor at the top has a multistrand cable weighing 4.5lb/ft. When the car is at the first floor, 180ft of cable are pulled out, and effectively 0ft are out when the car is at the top floor. How much work does the motor do just lifting the cable when it takes the car from the first floor to stop.
work = integral w dz
w = 4.5 (180-z)
so
work = integral from z = 0 to z = 180 of 4.5 (180-z)dz
= 4.5*180 (180) - 4.5 (1/2)(180)^2
= (1/2)(4.5)(180)(180)
or half the weight over the whole distance of course :)
To find the work done by the motor in lifting the cable, we need to calculate the change in potential energy of the cable.
The potential energy of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
First, let's calculate the mass of the cable. We are given that the cable weighs 4.5 lb/ft. Since 180 ft of cable are pulled out, the mass of this section of cable can be calculated as follows:
Mass of cable = 180 ft * (4.5 lb/ft) = 810 lb
Next, we need to determine the height that the cable is lifted. Since effectively 0 feet of cable are out when the car is at the top floor, the height is equal to the distance the car travels vertically. We are not given this distance, so we cannot calculate the exact work done. However, if we assume the elevator only goes vertically without any horizontal travel, the height will be 180 ft.
Now, we can calculate the change in potential energy (ΔPE) of the cable:
ΔPE = mgΔh
where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s²), and Δh is the change in height.
Converting the mass from pounds to kilograms:
Mass of cable = 810 lb * (0.4536 kg/lb) = 367.416 kg
Calculating the change in potential energy:
ΔPE = (367.416 kg) * (9.8 m/s²) * (180 ft * 0.3048 m/ft) = (367.416 kg) * (9.8 m/s²) * (54.864 m) = 187,657.96 J
Therefore, the work done by the motor in lifting the cable from the first floor to stop is approximately 187,657.96 Joules.
To calculate the work done in lifting the cable, we need to determine the weight of the cable that is lifted from the first floor to the top floor.
We know that the weight of the cable is 4.5 lb/ft, and 180 ft of cable are pulled out when the car is at the first floor.
So, the total weight of the cable that is lifted is given by:
Weight of cable = 4.5 lb/ft * 180 ft = 810 lb
Now, we can calculate the work done in lifting the cable using the formula:
Work = Force * Distance
In this case, the force is equal to the weight of the cable, which is 810 lb, and the distance is the height of the elevator shaft, which is the distance from the first floor to the top floor.
However, since we are only interested in the work done just lifting the cable, we need to subtract the work done when the car is also lifted. Since the car is not lifted by the motor pulling the cable, but by another mechanism, we can assume that the work done to lift the car is negligible in this calculation.
Therefore, the work done just lifting the cable when the car goes from the first floor to stop is given by:
Work = Force * Distance
= 810 lb * (distance from first floor to top floor)
It is important to note that the exact height of the elevator shaft or distance from the first floor to the top floor is not provided in the question. Hence, without this information, we cannot determine the exact value of the work done.