A fully loaded, slow-moving freight elevator has a cab with a total mass of 1250 kg, which is required to travel upward 58 m in 3.4 min, starting and ending at rest. The elevator's counterweight has a mass of only 1005 kg, and so the elevator motor must help. What average power is required of the force the motor exerts on the cab via the cable?
________ W
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 the motor exerts on the cab via the cable.
Work = Force × Distance
The force can be calculated using Newton's second law:
Force = Mass × Acceleration
The mass we need to consider is the total mass of the cab and its load, which is 1250 kg + 1005 kg = 2255 kg.
The acceleration can be calculated using the equations of motion:
Distance = Initial Velocity × Time + (1/2) × Acceleration × Time²
Since the initial velocity and final velocity are both zero, the equation simplifies to:
Distance = (1/2) × Acceleration × Time²
Rearranging the equation to solve for acceleration:
Acceleration = (2 × Distance) / (Time²)
Plugging in the given values:
Acceleration = (2 × 58 m) / (3.4 min * 60 s/min) = 0.096 m/s²
Now we can calculate the force:
Force = Mass × Acceleration = 2255 kg × 0.096 m/s² = 216.48 N
Next, let's calculate the work:
Work = Force × Distance = 216.48 N × 58 m = 12549.28 J
Finally, we can calculate the average power:
Average Power = Work / Time = 12549.28 J / (3.4 min * 60 s/min) = 102.27 W
Therefore, the average power required of the force the motor exerts on the cab via the cable is approximately 102.27 W.
To find the average power required of the force exerted by the motor on the cab, we need to use the formula:
Power = Work / Time
First, let's calculate the work done by the force exerted on the cab. The work done is equal to the change in potential energy of the cab.
The potential energy is given by the formula:
Potential Energy = Mass * Gravity * Height
Here,
Mass of the cab = 1250 kg
Height = 58 m
Gravity = 9.8 m/s^2 (approximate value on Earth)
So, the potential energy is:
Potential Energy = 1250 kg * 9.8 m/s^2 * 58 m
Next, we need to convert the time from minutes to seconds, as power is usually measured in Watts, which is joules per second. 1 minute is equal to 60 seconds.
Time = 3.4 min * 60 s/min
Now, we can calculate the average power:
Power = Potential Energy / Time
Substituting the values, we get:
Power = (1250 kg * 9.8 m/s^2 * 58 m) / (3.4 min * 60 s/min)
Simplifying the equation, we get the answer in Watts (W).