A helicopter winches up an injured person from a boat. The person plus equipment has a mass of 104 kg. If the height of the lift is 84 m and it is carried out at a uniform rate over 70 s, what is the power expended by the helicopter's winch motor? Assume that the engine is 61% efficient.
Since the load moves at a constant speed the force applied by the engine is equal to the gravity force. Therefore, the work is
W=F•h = m•g•h.
Effective output is
P = W/t= m•g•h/t
Power of the engine is P/0.61 = m•g•h/0.61• t=
104•9.8•84/0.61•70=2005 W =2.005 kW
To calculate the power expended by the helicopter's winch motor, we need to first calculate the work done by the winch.
The work done (W) is given by the formula:
W = force x distance
In this case, the force is the weight of the person plus equipment, which can be calculated using the formula:
Force = mass x gravity
where mass is the total mass of the person plus equipment (104 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
Force = 104 kg x 9.8 m/s^2
Next, we calculate the distance over which the work is done. In this case, it is the vertical distance the person is lifted, which is given as 84 m.
Now we can calculate the work done:
W = Force x distance
W = (104 kg x 9.8 m/s^2) x 84 m
The next step is to calculate the elapsed time. In this case, it is given as 70 seconds.
To calculate power, we use the formula:
Power = Work / Time
Let's calculate the work first:
W = (104 kg x 9.8 m/s^2) x 84 m
W = 81,055.2 Joules
Now we can calculate the power:
Power = Work / Time
Power = 81,055.2 J / 70 s
However, we also need to consider the efficiency of the winch motor, which is given as 61%. Efficiency is the ratio of useful work output to the amount of work input. In this case, the work input is the work done by the winch motor, and the useful work output is the work done in lifting the person plus equipment.
So, the actual work done by the winch motor is:
Actual work = Work / Efficiency
Actual work = 81,055.2 J / 0.61
Now we can calculate the power expended by the winch motor:
Power = Actual work / Time
Power = (81,055.2 J / 0.61) / 70 s
Calculating this expression will give us the power expended by the helicopter's winch motor.