A hydraulic lift uses liquid to transmit pressure from one piston to another.
The small piston usually receives the input force, and the large piston produces the output force.
1. Pascal’s principle
Pascal’s principle says that pressure applied to a confined liquid is transmitted equally throughout the liquid.
If two pistons are connected by the same liquid at the same level, then the pressure is the same:
\[
\begin{aligned}
P_1 &= P_2.
\end{aligned}
\]
Because pressure is force divided by area,
\[
\begin{aligned}
P &= \frac{F}{A}.
\end{aligned}
\]
Therefore, for two pistons,
\[
\begin{aligned}
\frac{F_1}{A_1}
&= \frac{F_2}{A_2}.
\end{aligned}
\]
2. Finding output force
If the input force \(F_1\), small piston area \(A_1\), and large piston area \(A_2\) are known, then
\[
\begin{aligned}
F_2
&= F_1\frac{A_2}{A_1}.
\end{aligned}
\]
This shows why a hydraulic lift can multiply force: if \(A_2\) is much larger than \(A_1\), then \(F_2\) is much larger than \(F_1\).
3. Finding required input force
If the output load \(F_2\) is known and we want the required input force, rearrange:
\[
\begin{aligned}
F_1
&= F_2\frac{A_1}{A_2}.
\end{aligned}
\]
A larger output piston area reduces the input force needed to lift the same load.
4. Mechanical advantage
Mechanical advantage compares output force to input force:
\[
\begin{aligned}
\mathrm{MA}
&= \frac{F_2}{F_1}.
\end{aligned}
\]
For an ideal hydraulic lift,
\[
\begin{aligned}
\mathrm{MA}
&= \frac{A_2}{A_1}.
\end{aligned}
\]
If the large piston area is \(30\) times the small piston area, then the ideal mechanical advantage is \(30\).
5. Efficiency
Real hydraulic machines lose some energy through friction, seal resistance, fluid viscosity, and small fluid compression.
A simple efficiency factor \(\eta\) can be used:
\[
\begin{aligned}
F_{2,\mathrm{useful}}
&= \eta F_{2,\mathrm{ideal}}.
\end{aligned}
\]
For an ideal system, \(\eta=1\), or \(100\%\).
For a real system, \(\eta<1\).
6. Piston motion and volume conservation
A hydraulic lift multiplies force, but it does not create energy.
If the small piston moves down by distance \(d_1\), it pushes a liquid volume
\[
\begin{aligned}
V_1 &= A_1d_1.
\end{aligned}
\]
The same liquid volume raises the large piston:
\[
\begin{aligned}
V_2 &= A_2d_2.
\end{aligned}
\]
For an incompressible fluid,
\[
\begin{aligned}
A_1d_1 &= A_2d_2.
\end{aligned}
\]
Therefore,
\[
\begin{aligned}
d_2
&= \frac{A_1}{A_2}d_1.
\end{aligned}
\]
If \(A_2>A_1\), the large piston moves a smaller distance than the small piston.
This is the trade-off: more force, less distance.
7. Worked example
Suppose a hydraulic lift has:
\[
\begin{aligned}
A_1 &= 0.004\ \mathrm{m^2},\\
A_2 &= 0.12\ \mathrm{m^2},\\
F_1 &= 2500\ \mathrm{N}.
\end{aligned}
\]
First find the pressure in the liquid:
\[
\begin{aligned}
P
&= \frac{F_1}{A_1}\\
&= \frac{2500}{0.004}\\
&= 625000\ \mathrm{Pa}.
\end{aligned}
\]
The same pressure acts on the large piston:
\[
\begin{aligned}
F_2
&= PA_2\\
&= (625000)(0.12)\\
&= 75000\ \mathrm{N}.
\end{aligned}
\]
The mechanical advantage is
\[
\begin{aligned}
\mathrm{MA}
&= \frac{F_2}{F_1}\\
&= \frac{75000}{2500}\\
&= 30.
\end{aligned}
\]
So a \(2500\ \mathrm{N}\) input force can produce a \(75000\ \mathrm{N}\) ideal output force.
8. Formula summary
| Goal |
Formula |
Meaning |
| Pressure |
\(P=F/A\) |
Force per unit area |
| Pascal relation |
\(F_1/A_1=F_2/A_2\) |
Same pressure in both pistons |
| Output force |
\(F_2=F_1(A_2/A_1)\) |
Find lift force from input force |
| Input force |
\(F_1=F_2(A_1/A_2)\) |
Find effort needed for a load |
| Mechanical advantage |
\(\mathrm{MA}=A_2/A_1\) |
Ideal force multiplication |
| Displacement relation |
\(A_1d_1=A_2d_2\) |
Large piston moves less when it gives more force |
9. Assumptions
- The liquid is confined and nearly incompressible.
- The pistons are compared at the same fluid level.
- The pressure is transmitted equally through the liquid.
- The ideal model ignores friction, leakage, trapped air, and piston weight.
- Efficiency can be used as a simple correction for real systems.
Key idea: a hydraulic lift multiplies force because the same pressure acts over a larger area on the output piston.
