Efficiency can be defined as the effectiveness of the simple machine, or the percentage of energy supplied (work supplied) that will actually be transferred to the object (useful work).
When a simple machine is used, part of the energy is transmitted from the user to the machine. The machine then transmits this energy to an object, which can then be moved. In theory, all the energy transmitted to a simple machine (work supplied) will be transmitted to the object (useful work). In reality, however, friction causes part of the energy to be transformed into heat, resulting in a loss of energy.
To determine the efficiency of a simple machine, use the following formula:
|R = \displaystyle \frac {W_{u}}{W_{s}} \times 100|
where
|R| represents the percentage efficiency |\small \text {(%) }|
|W_{u}| represents useful work |\small \text {(J)}|
|W_{s}| represents the work supplied |\small \text {(J)}|
Using the equations of work, the following formula can also be used:
|R = \displaystyle \frac {F_{r} \cdot \triangle x_{r}}{F_{d} \cdot \triangle x_{m}} \times 100|
where
|R| represents the percentage efficiency |\small \text {(%) }|
|F_{r}| represents the resistive force |\small \text {(N)}|
|\triangle x_{r}| represents the resistive displacement |\small \text {(m)}|
|F_{d}| represents driving work
|\small \text {(N)}|
|\triangle x_{d}| represents the driving displacement |\small \text {(m)}|
A force of |\small \text {75 N}| is applied to the handle of a winch crank to raise a bucket from the bottom of a well. The crank is turned so that the hand travels a distance of |\small \text {32 m}|. If the bucket weighs |\small \text {275 N}| and the well is |\small \text {8 m}| deep, what is the winch's efficiency?
Here's what's known about this problem.
||\begin{align} F_r &= 275 \: \text {N} & x_r &= 8 \: \text {m}\\
F_d &= 75 \: \text {N} & x_d &= 32 \: \text {m}\\ \end{align}||
||\begin{align} R = \displaystyle \frac {F_{r} \cdot \triangle x_{r}}{F_{d} \cdot \triangle x_{d}} \times 100
\quad \Rightarrow \quad
R&=\displaystyle \frac {275 \: \text {N} \cdot 8 \: \text {m}}{75 \: \text {N} \cdot 32 \: \text {m}} \times 100 \\
&= 92 \: \text {%} \end{align}||
In situations where several forces are applied, it is also possible to calculate the efficiency from the forces exerted on a single machine.
The theoretical driving force |(F_{d_{theo}})| represents the minimum driving force to be applied (without friction) to the simple machine.
The real driving force|(F_{d_{real}})| represents the driving force to be applied to the single machine, taking frictional forces into account.
The following formula can be used to calculate output from driving forces:
|R = \displaystyle \frac {F_{d_{theo}}}{F_{d_{real}}} \times 100|
where
|R| represents the percentage efficiency |\small \text {(%)}|
|F_{d_{theo}}| represents the theoretical driving force |\small \text {(N)}|
|F_{d_{real}}| represents the actual driving force |\small \text {(N)}|
When the mechanical advantages are known, the following formula can be used to calculate the efficiency:
|R = \displaystyle \frac {MA_{real}}{MA_{theo}} \times 100|
where
|R| represents the percentage efficiency |\small \text {(%)}|
|MA_{real}| represents the real mechanical advantage
|MA_{theo}| represents the theoretical mechanical advantage
The mechanical advantage |(MA)|, or mechanical gain |(MG)|, is the ratio between the magnitude of the resisting force and the magnitude of the driving force.
The mechanical advantage represents the efficiency of a simple machine: every simple machine has its own mechanical advantage. However, frictional forces will influence this mechanical advantage. This is why it is necessary to determine the theoretical mechanical advantage or the actual mechanical advantage.
The theoretical mechanical advantage |(MA_{theo})| is used when the simple machine is not subject to any friction.
To calculate the theoretical mechanical advantage, the following formula can be used if there is no friction:
|MA_{theo} = \displaystyle \frac {F_{r}}{F_{d}}|
where
|MA_{theo}| represents the theoretical mechanical advantage
|F_{r}| represents the resistive force |\small \text {(N)}|
|F_{d}| represents the driving force |\small \text {(N)}|
The following formula can be used regardless of whether there is friction or not:
|MA_{theo} = \displaystyle \frac {\triangle x_{d}}{\triangle x_{r}}|
where
|MA_{theo}| represents the theoretical mechanical advantage
|\triangle x_{d}| represents the motor displacement |\small \text {(m)}|
|\triangle x_{r}| represents the resistive displacement |\small \text {(m)}|
Thereal mechanical advantage |(MA_{real})| is used when considering frictional forces.
It should never be used to calculate displacements, which are not subject to friction.
To calculate the actual mechanical advantage, the following formula can be used regardless of whether there is friction or not:
|MA_{real} = \displaystyle \frac {F_{r}}{F_{d}}|
where
|MA_{real}| represents the real mechanical advantage
|F_{r}| represents the resistive force |\small \text {(N)}|
|F_{d}| represents the driving force |\small \text {(N)}|
What are the theoretical and actual mechanical advantages of the winch used in the example at the top of this sheet?
Since the efficiency of the winch is not |\small \text {100 %}| (it is |\small \text {92 %}|), the winch is subject to frictional forces. It will therefore be possible to calculate the actual mechanical advantage using the force ratio.
||\begin{align} F_r &= 275 \: \text {N} &F_d &= 75 \: \text {N} \end{align}||
||\begin{align} MA_{real} = \displaystyle \frac {F_{r}}{F_{d}}
\quad \Rightarrow \quad
MA_{real} &= \displaystyle \frac {275 \text { N}}{75 \text { N}} \\
&= 3.6 \end{align}||
To calculate the theoretical mechanical advantage, we have no choice but to calculate the displacement ratio.
||\begin{align} x_r &= 8 \: \text {m} &x_d &= 32\: \text {m} \end{align}||
||\begin{align} MA_{theo} = \frac {\triangle x_{d}}{\triangle x_{r}} \quad \Rightarrow \quad
MA_{theo} &= \displaystyle \frac {\text {32 m}}{\text {8 m}} \\
&= 4 \end{align}||