Question

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

Answer 1

Hint: Simple machine, any of several devices with few or no moving parts that are used to modify motion and force in order to perform work. The simple machines are the inclined plane, lever, wedge, wheel and axis pulley and screw.

Complete step by step answer:
Machine is a device which multiplies the force or multiplies the speed or it changes the direction of effort. There are some terms related to machines like load , effort, velocity ratio, mechanical advantage and efficiency.
Load is the output which is overcome by machine whereas the effort is the input to the machine.
Mechanical advantage is the ratio of load to effort of machine or we can say it is the ratio of output of machine to the input of the machine.
Velocity ratio is also defined as velocity of effort to the velocity of load.

Let a machine overcome a load ‘L’ by the application of an effort E. In time ‘t’. Let the displacement of effort of dE and displacement of load be \[{{d}_{1}}\]
Work input = \[\text{Effort }\times \text{displacement of effort}=E\times dE\]
Work output = \[\text{Load }\times \text{displacement of load}=\text{L}\times \text{dL}\]
Efficiency \[\left( \eta \right)=\dfrac{\text{Work output }}{\text{Work input}}\]
\[=\dfrac{L\times dL}{E\times dE}\text{ }.............\text{ 1}\]
Also, Mechanical Advantage, (M.A.) \[=\dfrac{\text{Load }\left( \text{L} \right)}{\text{Effort }\left( \text{E} \right)}\text{ }...........\text{ 2}\]
And velocity Ratio (V.R.) \[=\dfrac{dE}{-dL}\text{ }..........\text{ 3}\]
By equation 1,2 and 3
Efficiency \[\left( \eta \right)=\dfrac{\text{M}\text{.A}\text{.}}{\text{V}\text{.R}\text{.}}\]

\[\text{M}\text{.A}\text{.}=\text{V}\text{.R}\text{.}\times \eta \]
Thus, the mechanical advantage of the machine is equal to the product of its efficiency and Velocity ratio.

Note:
Mechanical Advantage (M.A.), efficiency \[\left( \eta \right)\] and velocity ratio are interconnected with each other. Basically mechanical Advantage of machine is equal to the product of its efficiency and velocity Ratio. Therefore,
\[\text{M}\text{.A}\text{.}=\text{V}\text{.R}\text{.}\times \eta \]