An elevator cab has a mass of 4500 kg and can carry a maximum load of 1800 kg. If the cab is moving upward at full load at 3.80 m/s, what power is required of the force moving the cab to maintain that speed?
The correct answer and explanation is :
To find the power required to move the elevator cab at a constant speed of 3.80 m/s while carrying its maximum load, we need to consider the total force needed to overcome gravitational forces and maintain that speed. Power is defined as the rate at which work is done, and mathematically it is given by:
[
P = F \cdot v
]
Where:
- (P) is the power,
- (F) is the total force exerted by the motor to move the cab, and
- (v) is the velocity of the cab.
Step 1: Determine the total mass being moved
The total mass of the elevator cab includes both the mass of the elevator itself and the mass of the load it is carrying. The mass of the cab is 4500 kg, and the maximum load is 1800 kg. Therefore, the total mass is:
[
m_{\text{total}} = m_{\text{cab}} + m_{\text{load}} = 4500 \, \text{kg} + 1800 \, \text{kg} = 6300 \, \text{kg}
]
Step 2: Calculate the force required to overcome gravity
The force required to counteract gravity is the weight of the elevator and its load. This force can be calculated using the formula:
[
F_{\text{gravity}} = m_{\text{total}} \cdot g
]
Where:
- (g) is the acceleration due to gravity, approximately 9.81 m/s².
Thus, the gravitational force is:
[
F_{\text{gravity}} = 6300 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 61,803 \, \text{N}
]
Step 3: Calculate the power required
Since the elevator is moving upward at a constant speed of 3.80 m/s, the motor must apply a force equal to the gravitational force to maintain that speed. The power required is then:
[
P = F_{\text{gravity}} \cdot v = 61,803 \, \text{N} \cdot 3.80 \, \text{m/s}
]
[
P = 234,849.4 \, \text{W} \, \text{or} \, 234.85 \, \text{kW}
]
Conclusion
The power required to move the elevator cab at a constant speed of 3.80 m/s while carrying a full load is approximately 234.85 kW.
Explanation
To keep the elevator moving at a constant velocity, the motor must exert a force equal in magnitude to the gravitational force pulling the elevator downward. The power is the rate at which the motor does work to overcome gravity. Since the elevator is moving at a constant speed, no additional force is required to accelerate it, but the motor must continuously apply enough force to counteract the downward gravitational pull. The calculated power reflects the energy the motor needs to maintain this motion without speeding up or slowing down.