Question

(a) State how the AMA and TMA of an inclined plane vary with the inclination of the plane. (b) State how the efficiency of an inclined plane varies with the inclination of the plane, and explain the reason for this variation.

Short answer

AMA, TMA, and efficiency all generally decrease as the inclination of an inclined plane increases due to increased friction and required input force.

Step-by-step solution

Understanding AMA and TMA

The Actual Mechanical Advantage (AMA) of an inclined plane is the ratio of the output force (weight of the object) to the input force applied. The Theoretical Mechanical Advantage (TMA) is the ratio of the distance moved by effort to the distance moved by the load. AMA and TMA can be expressed as \( \text{AMA} = \frac{\text{Output Force}}{\text{Input Force}} \) and \( \text{TMA} = \frac{\text{Length of Incline}}{\text{Height of Incline}} \) respectively.

Relation of TMA with Inclination

As the inclination of the plane increases, the length of the inclined plane decreases while the height remains constant, resulting in a decrease in TMA. Thus, TMA is inversely proportional to the inclination of the plane.

Relation of AMA with Inclination

The AMA is affected by friction and other resistive forces. As the inclination increases, the input force required to move the object increases due to higher gravitational component along the plane, thereby decreasing the AMA. Therefore, AMA generally decreases with increased inclination.

Efficiency of Inclined Plane

Efficiency is the ratio of AMA to TMA, expressed as \( \text{Efficiency} = \frac{\text{AMA}}{\text{TMA}} \times 100 \% \). As the incline increases, the gravitational component along the incline increases, requiring more input force and increasing influence of friction.

Variation of Efficiency with Inclination

The efficiency of an inclined plane decreases as the inclination increases. This happens because the increase in gravitational pull and friction at higher inclinations requires more input force relative to the output, reducing the actual efficiency. The smaller the angle, the less the friction and less input force needed, therefore efficiency is higher.

Key concepts

Actual Mechanical Advantage (AMA)

The Actual Mechanical Advantage (AMA) is a measure of how much an inclined plane helps to lift an object. It is calculated by dividing the output force, which is the weight of the object being lifted, by the input force, which is the force applied to move the object. This can be written as:

• \( \text{AMA} = \frac{\text{Output Force}}{\text{Input Force}} \)

Understanding the behavior of AMA as the inclination of the plane changes is essential. As the slope of the incline becomes steeper, you might assume the work becomes easier. However, as the incline increases, the input force needed also rises. This is due to the greater component of gravity pulling the object downwards on the steeper path. Consequently, the AMA tends to decrease when the inclination of the plane increases. Thus, efficiency in using the plane is reduced as more input force is needed to achieve the same lifting force.

Theoretical Mechanical Advantage (TMA)

The Theoretical Mechanical Advantage (TMA) differs from AMA because it assumes a frictionless and ideal scenario. TMA does not take into account losses due to friction or other resistive forces. It is calculated using the dimensions of the inclined plane:

• \( \text{TMA} = \frac{\text{Length of Incline}}{\text{Height of Incline}} \)

When the inclination of the plane is altered, TMA experiences change. Specifically, as the angle of the inclined plane increases, the length of the plane decreases, if height remains constant, causing the TMA to reduce. TMA is inversely proportional to the angle of inclination. So, in steeper inclines, even though you have to move the effort over a shorter board distance, you lose mechanical advantage. TMA helps us understand the potential efficiency of an inclined plane, but only in perfect conditions.

Efficiency of Inclined Plane

Efficiency is the comparison of the AMA over the TMA. It indicates how much of the input effort is successfully being converted into lifting effort without being lost to friction or other resistive forces. Efficiency is expressed as:

• \( \text{Efficiency} = \frac{\text{AMA}}{\text{TMA}} \times 100 \% \)

As the inclination of the plane is increased, the gravitational force pulling on the object along the plane becomes stronger, resulting in increased input force requirements. This situation is aggravated by friction, making the system less efficient as the incline becomes steeper. Thus, when the inclined plane is more horizontal, the efficiency is higher since less input force is lost to these resistive forces. The inclination plays a crucial role in determining how effectively the mechanical advantage augments the input force, translating to how many fewer resources are needed to achieve the intended movement.