Question

An elevator has mass 600 kg, not including passengers. the elevator is designed to ascend, at constant speed, a vertical distance of 20.0 m (five floors) in 16.0 s, and it is driven by a motor that can provide up to 40 hp to the elevator. what is the maximum number of passengers that can ride in the elevator? assume that an average passenger has mass 65.0 kg

Answer 1

Power = 40 hp ( 745.7 W / 1 hp)= 29828 W 
Power is the rate of energy so,
P = dW/dt = F(v) where F is force and v is velocity 
                 v = d/t = 20 m /16 s = 1.25 m/s 
                 F = P/v = 29828 W /1.25 m/s = 23862.4 N 
The force calculated is the force that is applied by the elevator to carry all of the weight (itself and the passengers). So,

Weight of the elevator = 600*9.81 = 5886N 
Weight of the passengers = 9.81m      where m is the mass of the passengers
 
For this system, in order to hold the maximum number of passengers it should be that the total force applied by the the elevator plus the passengers is equal to the force provided by the motor. Therefore, the elevator should be at rest or acceleration is 0 .

F - We - Wp = 0 
23862.4 N - 5886 N - 9.81m = 0 
m = 1832.5 kg  (total mass of the passengers )

No. of passengers = 1832.5 kg / 65 kg = 28.19

Thus, the maximum number of passenger is approximately 28.

Answer 2

28

Further explanation

Given:

• Mass of elevator = 600 kg, not including passengers.
• d = 20.0 m in t = 16.0 s, at constant speed.
• P = 40 horsepower.
• Assume that an average passenger has mass 65.0 kg.
• The acceleration of gravity, g = 9.8 m/s² on Earth.

Question:

What is the maximum number of passengers that can ride in the elevator?

The Process:

Step-1: convert hp to watts

For an approximate result, multiply the power value in horsepower by 746 .

$\boxed{ \ 40 \ hp \times 746 \ \frac{watts}{hp} = 29828 \ watts \ }$

Step-2: calculate the amount of work from the elevator

The relationship between power (P), work (W), and time (t) can be formulated as follows.

$\boxed{ \ P = \frac{W}{t} \ }$ $\rightarrow \boxed{ \ W = Pt \ }$

$\boxed{ \ W = (29828 \ watts)(16.0 \ seconds) \ }$

So, the work that can be carried out by the elevator is 477248 joules.

Step-3: calculate the net force

The relationship between work (W), the net force (∑F), and displacement (d) can be formulated as follows.

$\boxed{ \ W = (\Sigma F)(d) \ }$ $\rightarrow \boxed{ \ \Sigma F = \frac{W}{d} \ }$

$\boxed{ \ \Sigma F = \frac{477248 \ J}{20.0 \ m} \ }$

Thus, the net force of the elevator is 23862.4 N.

Final step: calculate the maximum number of passengers that can ride in the elevator

We prepare the weight of the elevator

$\boxed{ \ w = (600 \ kg)(9.8 \ m/s^2) = 5880 \ N \ }$

The net force of the elevator is the sum of the weight of the elevator and the passengers.

∑F = the weight of the lift + the weight of the passengers

23862.4 = 5880 + the weight of the passengers

The weight of the passengers = 17982.4 N.

Then we calculate the mass of the passengers.

$\boxed{ \ m = \frac{w}{g} \ } \rightarrow$ $\boxed{ \ \Sigma m = \frac{17982.4}{9.8} = 1834.94 \ kg \ }$

Finally, the mass of the passengers is divided by the average mass of a passenger.

$\boxed{ \ N = \frac{1834.94 \ kg}{65 \ kg/person} \ }$ $\boxed{ \ N \approx 28.23 \ }$

Thus, the maximum number of passengers that can ride in the elevator is approximately 28.

If 29 passengers enter the elevator right away, a small alarm signals an overload immediately ringing. The elevator will not move up.

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