Question

For a given machine (not ideal) prove:

\[\text{Efficiency}=\frac{\text{Actual mechanical advantage}}{\text {Ideal mechanical advantage}}\]

Answer 1

Let a useful load \[ l \] move through distance \[ d \] when effort \[ E \] moves through distance \[ D \].

\[ \text{Efficiency, } \eta = \frac{\text{useful output}}{\text{input}} = \frac{l \times d}{E \times D} = \frac{l/E}{D/d} \]

Here, \[ \frac{l}{E} \] = actual mechanical advantage (AMA), and \[ \frac{D}{d} \] = velocity ratio (VR). Therefore,

\[ \eta = \frac{\mathrm{AMA}}{\mathrm{VR}} \]

For a given machine, its ideal mechanical advantage (IMA) equals its velocity ratio (VR). Hence,

\[ \eta = \frac{\mathrm{AMA}}{\mathrm{IMA}} \]

Thus, \[ \text{Efficiency} = \frac{\text{Actual mechanical advantage}}{\text{Ideal mechanical advantage}} \]