The ideal mechanical advantage (IMA) of the pulley below is 2.0. The load on the pulley has a mass of 50 kg A worker applies a 280 N force on the rope and pulls 6.0 meters of rope. The mass is raised 30 meters at constant velocity. 16a. What is the efficiency of the pulley? 16b. What is the actual mechanical advantage (AMA) of this pulley?
The Correct Answer and Explanation is :
To determine the efficiency and actual mechanical advantage (AMA) of the pulley system, we’ll analyze the given data and apply relevant mechanical principles.
Given Data:
- Ideal Mechanical Advantage (IMA): 2.0
- Load mass: 50 kg
- Applied force: 280 N
- Rope pulled: 6.0 meters
- Load raised: 3.0 meters
16a. Efficiency of the Pulley System
Efficiency ((\eta)) is calculated using the formula:
[
\eta = \left( \frac{\text{AMA}}{\text{IMA}} \right) \times 100\%
]
First, we need to determine the Actual Mechanical Advantage (AMA).
16b. Actual Mechanical Advantage (AMA)
AMA is defined as the ratio of the output force (load) to the input force (effort):
[
\text{AMA} = \frac{\text{Output Force}}{\text{Input Force}}
]
The output force is the weight of the load, which can be calculated as:
[
\text{Output Force} = \text{mass} \times g = 50\,\text{kg} \times 9.8\,\text{m/s}^2 = 490\,\text{N}
]
Given the applied input force is 280 N:
[
\text{AMA} = \frac{490\,\text{N}}{280\,\text{N}} \approx 1.75
]
Now, substituting AMA and IMA into the efficiency formula:
[
\eta = \left( \frac{1.75}{2.0} \right) \times 100\% = 87.5\%
]
Explanation:
The Ideal Mechanical Advantage (IMA) of 2.0 suggests that, in a frictionless and perfectly efficient system, the input force required to lift the load would be half of the load’s weight. For a 50 kg mass (490 N), the ideal input force would be:
[
\text{Ideal Input Force} = \frac{490\,\text{N}}{2} = 245\,\text{N}
]
However, the worker applies a force of 280 N, indicating the presence of inefficiencies such as friction within the pulley system. The Actual Mechanical Advantage (AMA) of 1.75 reflects these real-world factors, showing that the system doesn’t achieve the ideal performance.
Efficiency quantifies how effectively the pulley system converts input work into useful output work. An efficiency of 87.5% means that 87.5% of the input energy is used to lift the load, while the remaining 12.5% is lost due to factors like friction.
It’s also noteworthy that the worker pulls 6.0 meters of rope to raise the load by 3.0 meters. This 2:1 ratio aligns with the IMA of 2.0, as the distance the effort moves is twice the distance the load moves in an ideal system. However, due to real-world inefficiencies, the actual force required is slightly higher than the ideal calculation, resulting in an AMA less than the IMA.
Understanding these distinctions between IMA, AMA, and efficiency is crucial for designing and utilizing pulley systems effectively, as it allows for the anticipation of energy losses and the adjustment of input forces to achieve desired outcomes.