Question

Write an expression to show the relationship between mechanical advantage, velocity ratio, and efficiency for a simple machine.

Answer 1

Step 1: Defining mechanical advantage, velocity ratio and efficiency,

Mechanical Advantage $\left(\mathrm{MA}\right)$:

1. It is the ratio of the load to the effort.
2. It is expressed as, $\mathrm{MA}=\frac{\mathrm{Load}\left(L\right)}{\mathrm{Effort}\left(E\right)}$.

Velocity Ratio $\left(\mathrm{VR}\right)$:

1. It is defined as the ratio of the velocity of the effort to the velocity of the load.
2. It is expressed as, $\mathrm{VR}=\frac{\text{Distance moved by effort}\left({\text{d}}_E\right)}{\text{Distance moved by load}\left({\text{d}}_L\right)}$.

Efficiency $\left(\eta \right)$:

1. It is defined as the ratio of work done $\left(W_{\text{output}}\right)$ by the machine to work done $\left(W_{\text{input}}\right)$ on the machine.
2. It is expressed as $\eta =\frac{\text{Work done}\text{by the machine}}{\text{Work done on the machine}}=\frac{W_{\text{output}}}{W_{\text{input}}}$.

Step 2: Efficiency in terms of mechanical advantage and velocity ratio is,

An expression to show the relationship between mechanical advantage, velocity ratio, and efficiency for a simple machine is shown below,

As the work done by the machine in terms of load and distance traveled by the load is,

$W_{output}=L\times d_L$

Similarly, work done on the machine is,

$W_{input}=E\times d_E$

We know, $\mathrm{MA}=\frac{L}{E}$, $\mathrm{VR}=\frac{d_E}{d_L}$ and $\eta =\frac{W_{\text{output}}}{W_{\text{input}}}$

Thus,

$$\begin{gathered} \eta =\frac{L\times d_L}{E\times d_E}\times 100\% \\ \eta =\frac{\frac{L}{E}}{{\displaystyle \frac{d_E}{d_L}}}\times 100\% \\ \eta =\frac{\mathrm{MA}}{\mathrm{VR}}\times 100\% \end{gathered}$$

Hence, the relationship between mechanical advantage, velocity ratio, and efficiency is $\eta =\frac{\mathrm{MA}}{\mathrm{VR}}\times 100\%$.