A fully loaded, slow-moving freight elevator has a cab with a total mass of 1800 kg, which is required to travel upward 50 m in 2.6 min, starting and ending at rest. The elevator's counterweight has a mass of only 780 kg, so the elevator motor must help pull the cab upward. What average power is required of the force the motor exerts on the cab via the cable?
average velocity = 50m/156 s
= 0.3205 m/s
average force required = (1800 - 780)kg*9.81 m/s^2 = 10,006 N
Power (average) = force x velocity = 3207 W = 4.3 horsepower
The power will need to be higher during acceleration, and lower during deceleration, but 4.3 Hp is the average needed.
To find the average power required of the force the motor exerts on the cab via the cable, we can use the formula:
Average Power = Work / Time
First, let's calculate the work done by the force exerted by the motor on the cab.
The work done is given by the formula:
Work = force * distance
In this case, the force is the net force acting on the cab, which is equal to the force exerted by the motor minus the force exerted by gravity. The force exerted by the motor can be calculated using Newton's second law:
Force = mass * acceleration
Since the elevator is moving upward, the acceleration is positive and given by:
acceleration = (final velocity - initial velocity) / time
The initial velocity is zero because the cab starts at rest. The final velocity can be calculated using the equation of motion:
final velocity = initial velocity + acceleration * time
The distance traveled is given as 50 m.
Now, let's calculate the force exerted by gravity on the cab. The force exerted by gravity is given by the formula:
Force_gravity = mass_cab * g
where g is the acceleration due to gravity.
Finally, we can substitute all these values into the formulas to calculate the average power.
Let's do the calculations step by step:
1. Calculate the force exerted by gravity on the cab:
Force_gravity = 1800 kg * 9.8 m/s^2 = 17640 N
2. Calculate the acceleration of the cab:
For this, we need to calculate the final velocity first:
final velocity = 0 + (50 m / 2.6 min) * (1 min / 60 s) = 0 + (50 m / 2.6 min) * (1 min / 60 s) = 2.884 m/s
Now, calculate the acceleration:
acceleration = (2.884 m/s - 0) / 2.6 min = 1.109 m/s^2
3. Calculate the force exerted by the motor on the cab:
Force_motor = mass_cab * acceleration = 1800 kg * 1.109 m/s^2 = 1996.2 N
4. Calculate the work done by the force exerted by the motor:
Work = Force_motor * distance = 1996.2 N * 50 m = 99810 J
5. Calculate the average power:
Average Power = Work / Time = 99810 J / (2.6 min * 60 s/min) = 637.69 W
Therefore, the average power required of the force the motor exerts on the cab is approximately 637.69 Watts.