Question

Derive the relation between mechanical advantage and velocity ratio and efficiency of a machine?

Answer 1

Mechanical advantage and its SI unit:

1. Mechanical advantage is a parameter that describes how much of the effort that is put into a simple machine, go into manipulating the load.
2. In other words, it is the ratio of the output work to the input work of a machine.
3. It can also be stated as the ratio of the load to the effort.

The formula of mechanical advantage(M.A.):

1. $Mechanicaladvantage(M.A)=\frac{Load\left(L\right)}{Effort\left(E\right)}$
2. This ratio is a measure of the advantage that one obtains by using the machine.
3. Since it is the ratio of two identical quantities, it has no units.

Velocity ratio(V.R):

1. It is the ratio of the velocity at which the effort is applied to the machine to the velocity at which the load moves.
2. It is also defined as the ratio of displacement of effort to the displacement to load.

The formula for velocity ratio:

$$\begin{gathered} Velocityratio(V.R)=\frac{Velocityofeffort\left(V_E\right)}{Velocityofload\left(V_L\right)} \\ If,d_E=displacementcausedbyeffortintimet \\ {d}_L=displacementcausedbyloadinsametimet \\ then, \\ Velocityeffort=\frac{d_E}{t} \\ Velocityload=\frac{d_L}{t} \\ Therefore, \\ Velocityratio=\frac{d_{E/t}}{d_L/t}=\frac{d_E}{d_L} \end{gathered}$$

2. Since velocity ratio is the ratio of two similar quantities, hence it has no unit.

Relation between the mechanical advantage (MA) and velocity ratio (VR) of an ideal machine:

1. $Effeciency=Mechanicaladvantage/velocityratio$
2. For the ideal machine $Effeciency=1$
3. So, for the ideal machine, $Mechanicaladvantage=velocityratio$.

Efficiency:

1. The efficiency of a machine is the ratio of the work done on the load by the machine to the work done on the machine by the effort.
2. Thus, it is the ratio of useful work done by the machine output to the work done by the machine input.
3. It is represented by the Greek symbol$\eta \left(eta\right)$.

The formula of efficiency:

1. $\eta =\frac{Output}{Inout}$
2. It is usually expressed as a percentage $\eta =\frac{Output}{Inout}\times 100\%$
3. Since it is a ratio of two similar quantities, it has no unit.

The relation between mechanical advantage and velocity ratio and efficiency of a machine:

1. If an effort E causes a displacement in time t, and a load L causes displacement in the same time, then

$$\begin{gathered} Input=Effort\times Displacementofeffort \\ {W}_{input}=E\times d_E \\ Output=Load\times Displacementofload \\ {W}_{output}=L\times d_L \\ Efficiency\left(\eta \right)=\frac{Output\left(W_{output}\right)}{Input\left(W_{input}\right)} \\ \eta =\frac{L\times d_L}{E\times d_E}=\frac{L}{E}\times \frac{d_L}{d_E} \\ Since, \\ \frac{L}{E}=MA \\ \frac{d_E}{d_L}=VR \\ \eta =\frac{MA}{VR} \\ Therefore, \\ MA=VR\times \eta \end{gathered}$$

For ideal machine:

1. For an ideal machine,

$$\begin{gathered} MA=VR \\ Thus,\eta =1 \end{gathered}$$

For actual machine :

1. For an actual machine,

$$\begin{gathered} MA<VR \\ Thus,\eta <1 \end{gathered}$$

Hence, efficiency is the ratio of mechanical advantage to velocity ratio of a machine