Question
Derive the relationship between mechanical advantage, velocity ratio and efficiency of a machine.
✓
Answer
Let a machine overcome a load L by the application of an effort E. In time $t$, let the displacement
of effort be $dE$ and the displacement of load be dL.
Work input $=$ Effort X displacement of effort
$= EXdE$
Efficiency $n=\frac{\text { work output }}{\text { work input }}$
$
\begin{aligned}
& n =\frac{L \times d L}{E \times d E}=\frac{L}{E} \times \frac{1}{d E / d L} \\
& \text { But } \frac{L}{E}=M \cdot A \\
& \frac{d E}{d L}=V \cdot R \\
& n =\frac{M \cdot A}{V \cdot R} \\
& \text { M.A }= n \times V \cdot R
\end{aligned}
$
Thus, mechanical advantage of a machine is equal to the product of its efficiency and velocity ratio.
of effort be $dE$ and the displacement of load be dL.
Work input $=$ Effort X displacement of effort
$= EXdE$
Efficiency $n=\frac{\text { work output }}{\text { work input }}$
$
\begin{aligned}
& n =\frac{L \times d L}{E \times d E}=\frac{L}{E} \times \frac{1}{d E / d L} \\
& \text { But } \frac{L}{E}=M \cdot A \\
& \frac{d E}{d L}=V \cdot R \\
& n =\frac{M \cdot A}{V \cdot R} \\
& \text { M.A }= n \times V \cdot R
\end{aligned}
$
Thus, mechanical advantage of a machine is equal to the product of its efficiency and velocity ratio.
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