An elevator cab in the New York Marriott Marquis has a mass of 4.96 x 103 kg and can carry a maximum load of 1.80 x 103 kg. If the cab is moving upward at full load at 3.71 m/s, what power is required of the force moving the cab to maintain that speed?
Add the elevator mass and the load mass. Multiply the sum by g to get the cable tension in Newtons. The product of cable tension and velocity is the power required, in Watts. If it were accleerating, still more power would be needed.
To calculate the power required, we need to determine the net force acting on the elevator cab and then multiply it by the velocity.
1. Calculate the net force:
The net force on the elevator cab is the sum of the gravitational force and the force required to maintain the upward velocity.
a) Gravitational force:
The gravitational force (weight) acting on the elevator cab is given by the formula:
F_gravity = m_total * g
where m_total is the total mass (mass of the cab + maximum load) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
m_total = mass of the cab + maximum load = 4.96 x 10^3 kg + 1.80 x 10^3 kg = 6.76 x 10^3 kg
F_gravity = (6.76 x 10^3 kg) * (9.8 m/s^2) = 6.6288 x 10^4 N
b) Force required to maintain upward velocity:
The force required to maintain the upward velocity can be calculated using Newton's second law:
F_velocity = m_total * a
where a is the acceleration. In this case, since the cab is moving upward at a constant speed, the acceleration is zero.
F_velocity = (6.76 x 10^3 kg) * 0 = 0 N
The net force is the sum of the gravitational force and the force required to maintain upward velocity:
Net force = F_gravity + F_velocity = 6.6288 x 10^4 N + 0 N = 6.6288 x 10^4 N
2. Calculate the power:
Power is given by the formula:
Power = force * velocity
Power = (6.6288 x 10^4 N) * (3.71 m/s) = 2.4569 x 10^5 W
Therefore, the power required to maintain the speed of the elevator cab at full load is 2.4569 x 10^5 W.
To find the power required to maintain the speed of the elevator cab, we need to calculate the force exerted by the elevator cab and then multiply it by the cab's velocity.
First, let's find the force exerted by the cab. Since the cab is moving upward at full load, the force exerted by the cab is equal to its weight plus the weight of the load it is carrying.
The weight of the cab can be calculated using the formula:
Weight = mass * acceleration due to gravity
Given that the mass of the cab is 4.96 x 10^3 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate the weight of the cab as follows:
Weight of the cab = 4.96 x 10^3 kg * 9.8 m/s^2
Next, let's calculate the weight of the load. The load's weight can also be calculated using the formula:
Weight = mass * acceleration due to gravity
Given that the maximum load the cab can carry is 1.80 x 10^3 kg, we can calculate the weight of the load as follows:
Weight of the load = 1.80 x 10^3 kg * 9.8 m/s^2
Now, let's find the total force exerted by the cab. Since the elevator is moving upward at a constant velocity, the total force exerted by the cab is equal to the sum of the weight of the cab and the weight of the load it is carrying:
Total force = Weight of the cab + Weight of the load
Finally, let's calculate the power required using the formula:
Power = Force * Velocity
Plug in the value of the total force and the velocity of the cab to find the power required.
This will give you the power required of the force moving the cab to maintain that speed.