Problem 9 - Applied Physics

Question

If the efficiency of a pulley is \(82 \%\) and its IMA is 22 , what is its AMA?

Step-by-Step Solution

Answer

The AMA of the pulley is 18.04.

Step 1: Understanding the Given Values

In the problem, we are given the efficiency of the pulley system as 82% and the Ideal Mechanical Advantage (IMA) as 22. We need to find the Actual Mechanical Advantage (AMA).

Step 2: Formula for Efficiency

The efficiency of the pulley system is given by the formula: \[ \text{Efficiency} = \left( \frac{\text{AMA}}{\text{IMA}} \right) \times 100 \% \] We need to rearrange this formula to solve for AMA.

Step 3: Rearranging the Efficiency Formula

To find AMA, rearrange the formula:\[ \frac{\text{AMA}}{\text{IMA}} = \frac{\text{Efficiency}}{100 \%} \]Thus,\[ \text{AMA} = \text{IMA} \times \frac{\text{Efficiency}}{100 \%} \]

Step 4: Substituting the Given Values

Substitute the known values (IMA = 22 and Efficiency = 82%) into the rearranged formula:\[ \text{AMA} = 22 \times \frac{82}{100} \]

Step 5: Calculating the AMA

Perform the multiplication to calculate AMA:\[ \text{AMA} = 22 \times 0.82 = 18.04 \]Thus, the Actual Mechanical Advantage is 18.04.

Key Concepts

Efficiency of Pulley Systems

Efficiency in pulley systems is an important metric that shows how well the system converts input energy into useful work. It is a percentage measure that compares the Actual Mechanical Advantage (AMA) to the Ideal Mechanical Advantage (IMA).
The efficiency can be computed using the formula:

• \[ \text{Efficiency} = \left( \frac{\text{AMA}}{\text{IMA}} \right) \times 100 \%\]

This implies that if a pulley system is perfectly efficient, its efficiency would be 100%. However, real-life pulley systems, due to friction and other factors, tend to have efficiencies less than 100%. Understanding efficiency helps in optimizing pulley systems for various tasks.

Mechanical Advantage

Mechanical Advantage (MA) is a fundamental concept in physics that reflects how much a machine can multiply force. It is the ratio of the output force exerted by a machine to the input force applied to it.
Mechanical Advantage can help us understand how machines, like pulley systems, make it easier to lift heavy loads by comparing the input and output forces:

• \[ \text{Mechanical Advantage (MA)} = \frac{\text{Output Force}}{\text{Input Force}}\]

A higher MA means less force is needed to do the same work, but it does not account for losses due to friction or other inefficiencies.

Actual Mechanical Advantage (AMA)

Actual Mechanical Advantage (AMA) considers the real-world conditions of a machine, including losses due to friction and other inefficiencies. It is the practical measure of how effectively a machine amplifies input force. Unlike Ideal Mechanical Advantage (IMA), AMA provides a more realistic picture of performance by reflecting these inevitable losses.
AMA is calculated using the ratio of the output force to the actual input force applied:

• \[ \text{AMA} = \frac{\text{Actual Output Force}}{\text{Actual Input Force}}\]

An example calculation shows that, if the efficiency of a pulley system is \(82\%\) and the IMA is \(22\), then the AMA is \(18.04\) when calculated using the efficiency formula.

Ideal Mechanical Advantage (IMA)

Ideal Mechanical Advantage (IMA) is a theoretical measure of a machine's efficiency without accounting for friction or any other real-world factors. It serves as an understanding of how a machine should perform under ideal conditions.
IMA is calculated by examining the geometry or design of a system, as it is the ratio of input distance to output distance:

• \[ \text{IMA} = \frac{\text{Input Distance}}{\text{Output Distance}}\]

The IMA of a pulley system can help us understand the potential performance in perfect circumstances, but it’s important to note that it usually overestimates the true efficiency due to its exclusion of real-world inefficiencies.