Efficiency can be defined as the effectiveness of the simple machine, or the percentage of energy supplied (work input) that will actually be transferred to the object (work output).
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 input) will be transmitted to the object (work output). 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:
|\text{Energy efficiency} = \displaystyle \frac {W_{out}}{W_{in}} \times 100|
where
|\text{Energy efficiency}| represents the percentage efficiency |\small \text {(%) }|
|W_{out}| represents work output |\small \text {(J)}|
|W_{int}| represents the work input|\small \text {(J)}|
Using the equations of work, the following formula can also be used:
|\text{Energy efficiency} = \displaystyle \frac {F_{r} \cdot \Delta x_{r}}{F_{d} \cdot \Delta x_{d}} \times 100|
where
|\text{Energy efficiency}| 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 force |\small \text {(N)}|
|\Delta x_{d}| represents the displacement caused by the driving force |\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 is displaced over |\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 energy 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} \text{Energy efficiency} = \displaystyle \frac {F_{r} \cdot \Delta x_{r}}{F_{d} \cdot \Delta x_{d}} \times 100
\quad \Rightarrow \quad
\text{Energy efficiency}&=\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 energy 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:
|\text{Energy efficiency} = \displaystyle \frac {F_{d_{theo}}}{F_{d_{real}}} \times 100|
where
|\text{Energy efficiency}| represents the percentage efficiency |\small \text {(%)}|
|F_{d_{theo}}| represents the theoretical driving force |\small \text {(N)}|
|F_{d_{real}}| represents the real driving force |\small \text {(N)}|
When the mechanical advantages are known, the following formula can be used to calculate the energy efficiency:
|\text{Energy efficiency} = \displaystyle \frac {MA_{real}}{MA_{theo}} \times 100|
where
|\text{Energy efficiency}| 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 real 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 {\Delta x_{d}}{\Delta x_{r}}|
where
|MA_{theo}| represents the theoretical mechanical advantage
|\Delta x_{d}| represents the displacement caused by the driving force|\small \text {(m)}|
|\Delta x_{r}| represents the displacement caused by the resistive force |\small \text {(m)}|
The real 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 concept sheet?
Since the energy 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 {\Delta x_{d}}{\Delta x_{r}} \quad \Rightarrow \quad
MA_{theo} &= \displaystyle \frac {\text {32 m}}{\text {8 m}} \\
&= 4 \end{align}||