Question

Derive a relationship between mechanical advantage, velocity ratio and efficiency of a simple machine.

Answer 1

Let a machine overcome a load $L$ by the application of an effort $E$. In time $t$, let the displacement of effort of $d_E$ ad the displacement of load be $d_L$.
Work input $=Effort\times$ displacement of effort
$=E\times d_E$
Work output $=Load\times$ displacement of load
$=L\times d_L$
$Efficiency=\frac{\text{Work output}}{\text{Work input}}$
$=\frac{L\times d_L}{E\times d_E}=\frac{L}{E}\times \frac{d_L}{d_E}$
$=\frac{L}{E}\times \frac{1}{d_E/d_L}$
But $\frac{Load\left(L\right)}{Effort\left(E\right)}=M.A.$
and $\frac{d_E}{d_L}=V.R.$
$\therefore Efficiency\eta =\frac{M.A.}{V.R.}$ or $M.A.=V.R.\times \eta$
Thus, the mechanical advantage of a machine is equal to the product of its efficiency and Velocity Ratio.