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 solve this exercise, we need to follow a step-by-step approach. Here's how we can solve it:
Step 1: Calculate the input power to the gearbox in Watts.
Given: Power input to the gearbox = 5 kW
Multiplying by 1000 to convert kW to Watts, we get:
Power input to the gearbox = 5,000 Watts
Step 2: Calculate the output power from the gearbox.
Output power = Efficiency * Input power
Given: Efficiency = 85% = 0.85 (in decimal form)
Output power = 0.85 * 5,000 = 4,250 Watts
Step 3: Convert the output power from Watts to Torque.
We can use the formula:
Torque (in Nm) = Power (in Watts) / (2π * Speed (in RPM) / 60)
Rearranging the formula, we have:
Speed (in RPM) = (Power (in Watts) / (2π * Torque (in Nm) / 60)
Step 4: Calculate the speed at which the load may be lifted.
Given: Power output from the gearbox = 4,250 Watts
Pitch diameter of the winch drum = 220 mm = 0.22 m (converting mm to m)
Using the above formula, we can solve for the speed:
Speed = (4,250 / (2π * Torque / 60))
Step 5: Find the maximum load capacity of the winch.
Given: Reduction ratio of the gearbox = 50:1
We know that torque is inversely proportional to the reduction ratio.
So, Torque = (Speed * Reducing ratio * 2π * Pitch diameter) / 60
Substituting the values, we can calculate the torque.
Step 6: Convert torque to load capacity.
To convert torque to load capacity, we need to consider the radius of the winch drum and the gravitational constant. Let's assume g = 9.81 m/s^2.
Load capacity (in N) = Torque (in Nm) * Generator ratio / Radius
Step 7: Convert load capacity to tonnes.
Given that 1 tonne = 9,810 N, we can calculate the maximum load capacity in tonnes.
Let's calculate the values step by step and find the answers for maximum speed and maximum load capacity:
Step 1: Power input to the gearbox = 5,000 Watts
Step 2: Output power from the gearbox = 0.85 (efficiency) * 5,000 (input power) = 4,250 Watts
Step 3: Speed (in RPM) = 4,250 / (2π * Torque (in Nm) / 60)
Step 4: Speed (in RPM) = 4,250 / (2π * 0.22 * Torque (in Nm) / 60)
Step 5: Torque (in Nm) = (Speed * 50 * 2π * 0.22) / 60
Step 6: Load capacity (in N) = Torque (in Nm) * 50 / (0.22 / 2)
Step 7: Maximum load capacity (in tonnes) = Load capacity (in N) / 9,810
By substituting the values into the equations and following the steps, you can find the maximum speed and maximum load capacity of the winch.