An elevator motor provides 45.0 kW of power while lifting a 2000 kg elevator 35 m. If the elevator contains six passengers of the same mass and it takes 20.0 s to accomplish this task, determine the mass of each passenger. Assume the system is 83 % efficient.
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To find the mass of each passenger, we need to first determine the work done by the elevator motor. Then, using the efficiency of the system, we can determine the total work done on the elevator and passengers. Finally, we can divide the total work by the distance traveled to find the mass of each passenger.
First, let's calculate the work done by the elevator motor.
The power provided by the motor is given as 45.0 kW. Power is defined as the rate at which work is done, so we will convert this to joules per second (J/s).
1 kW = 1000 J/s, so 45.0 kW = 45,000 J/s.
Since power is the rate at which work is done, we can find the work done by the motor by multiplying the power by the time:
work = power * time
= 45,000 J/s * 20.0 s
= 900,000 J
Now, let's calculate the total work done on the elevator and passengers by the system. Since the system is 83% efficient, the output work is 83% of the input work.
output work = efficiency * input work
= 0.83 * 900,000 J
= 747,000 J
Next, we can calculate the total mass being lifted. This includes the mass of the elevator and passengers, which is given as 2000 kg. Let's denote the mass of each passenger as m.
total mass = mass of elevator + mass of passengers
= 2000 kg + 6m
Now, using the equation for work, we have:
total work = force * distance,
where force = mass * acceleration due to gravity, and distance is given as 35 m.
total work = (total mass * acceleration due to gravity) * distance
= ((2000 kg + 6m) * 9.8 m/s^2) * 35 m
= (19600 kg * m + 2000 kg * 9.8 m/s^2 * 35 m
Since the total work is equal to the output work calculated earlier (747,000 J), we can equate the two expressions and solve for m:
747,000 J = (19600 kg * m + 2000 kg * 9.8 m/s^2 * 35 m
Simplifying this equation will give us the value of m, which represents the mass of each passenger.