Maximum Mechanical Advantage of a Lifting Machine and Friction

A lifting machine is used to raise a heavy load (W) by applying a smaller effort (P). The performance of such a machine is evaluated using its Mechanical Advantage (MA).

The Mechanical Advantage (MA) is the ratio of the load lifted to the effort applied:

\( MA = \frac{W}{P} \)

For any real lifting machine, there are always losses due to friction. The relationship between the effort (P) required to lift a load (W) is often described by a linear equation known as the Law of the Lifting Machine:

\( P = aW + b \)

In this equation:

• P represents the effort applied.
• W represents the load being lifted.
• 'a' is a constant that accounts for the effort required to overcome friction which is proportional to the load, and also related to the ideal mechanical advantage (velocity ratio).
• 'b' is a constant that accounts for the effort required to overcome friction that is independent of the load (like friction in bearings when the machine is unloaded).

To find the Mechanical Advantage using the law of the machine, we substitute the expression for P into the MA formula:

\( MA = \frac{W}{aW + b} \)

The mechanical advantage of a real machine is not constant; it generally increases as the load W increases. The maximum mechanical advantage is achieved under a very large load, ideally when the load W approaches infinity. At very high loads, the constant friction term 'b' becomes negligible compared to the load-dependent term 'aW'.

Let's find the limit of the MA as W tends towards infinity:

\( \text{Maximum } MA = \lim_{W \to \infty} \frac{W}{aW + b} \)

To evaluate this limit, we can divide both the numerator and the denominator by W:

\( \text{Maximum } MA = \lim_{W \to \infty} \frac{W/W}{(aW + b)/W} = \lim_{W \to \infty} \frac{1}{a + b/W} \)

As W becomes infinitely large, the term \( b/W \) becomes very small and approaches zero.

\( \text{Maximum } MA = \frac{1}{a + 0} = \frac{1}{a} \)

The question states that 'm is a constant called coefficient of friction'. In the context of the linear law \( P = aW + b \), the constant 'a' represents the effort needed per unit load at very high loads, which includes the effect of friction proportional to the load. If we identify this constant 'a' with the given coefficient 'm', then the maximum mechanical advantage is given by \( 1/m \).

Therefore, based on the interpretation of 'm' as the constant 'a' from the linear law of a lifting machine, the maximum mechanical advantage is \( 1/m \).