A motorised gearbox has a reduction ratio of 50:1. The power input to the gearbox is 5 kW at 1440 RPM. The gearbox efficiency is 85% under maximum load conditions. The output shaft of the gearbox is attached to a winch drum of pitch diameter 220 mm and is used to hoist a load.
Determine the maximum speed at which a load may be lifted and the maximum load capacity (in tonnes) of the winch. Frictional losses in the winch drum itself may be neglected.
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To determine the maximum speed at which a load may be lifted and the maximum load capacity of the winch, we'll need to consider several factors.
1. Calculate the input power going into the gearbox:
Power input = 5 kW
2. Determine the output power of the gearbox:
Output power = Efficiency * Power input
Output power = 0.85 * 5 kW
Output power = 4.25 kW
3. Calculate the output rotational speed of the gearbox:
Output speed = Input speed / Reduction ratio
Output speed = 1440 RPM / 50
Output speed = 28.8 RPM
4. Convert the output rotational speed to radians per second:
Output speed (radians/sec) = Output speed (RPM) * (2π / 60)
Output speed (radians/sec) = 28.8 * (2π / 60)
Output speed (radians/sec) ≈ 3.023 rad/s
5. Calculate the torque output of the gearbox:
Torque output = (Output power * 1000) / Output speed (radians/sec)
Torque output = (4.25 kW * 1000) / 3.023 rad/s
Torque output ≈ 1406.72 Nm
6. Calculate the force at the winch drum:
Force = Torque output / (Radius of drum)
Radius of drum = Diameter of drum / 2 = 220 mm / 2 = 0.11 m
Force = 1406.72 Nm / 0.11 m
Force ≈ 12,788 N
7. Determine the maximum speed of load lifting:
Maximum speed = Output speed * (2π / Reduction ratio)
Maximum speed = 28.8 RPM * (2π / 50)
Maximum speed ≈ 3.62 RPM
8. Convert the maximum speed from RPM to m/s:
Maximum speed (m/s) = Maximum speed (RPM) * (π / 30)
Maximum speed (m/s) ≈ 3.62 * (π / 30)
Maximum speed (m/s) ≈ 0.379 m/s
9. Finally, let's calculate the maximum load capacity of the winch in tonnes:
Maximum load capacity = Force / (gravitational constant * 1000)
gravitational constant = 9.81 m/s^2
Maximum load capacity ≈ 12,788 N / (9.81 m/s^2 * 1000)
Maximum load capacity ≈ 1.31 tonnes
Therefore, the maximum speed at which a load may be lifted is approximately 0.379 m/s, and the maximum load capacity of the winch is approximately 1.31 tonnes.