Pressure in Fluids
4. Pascal's Principle
Learning outcomes
- I can state Pascal's Principle in my own words.
- I can explain how pressure is transmitted through a fluid.
- I can describe how hydraulic systems multiply force.
- I can solve simple problems involving hydraulic devices.
- I can identify applications of Pascal's Principle in technology.
Pascal's Principle explains how pressure can be transmitted through a confined fluid. It is the scientific idea behind many hydraulic systems, including car brakes, hydraulic lifts, construction equipment, and industrial machinery.
One of its most useful applications is that a relatively small input force can be used to produce a much larger output force.
What Is Pascal's Principle?
Pascal's Principle states that a change in pressure applied to a confined fluid is transmitted equally throughout the fluid.
In simpler words:
If you increase the pressure at one point in an enclosed fluid, that pressure increase is transmitted throughout the fluid.
The principle is named after the French mathematician and scientist Blaise Pascal.
A fluid can be a liquid or a gas, but hydraulic systems normally use liquids because liquids are very difficult to compress.
Pressure in a Confined Liquid
Imagine a completely filled container of liquid with movable pistons at each end.
If you push down on one piston, you apply a force to the liquid.
Because:
Pressure = Force ÷ Area
the force produces pressure in the liquid.
That pressure is transmitted through the liquid to the other piston.
For an ideal hydraulic system:
Pressure at input = Pressure at output
or:
P₁ = P₂
This simple relationship makes hydraulic machines possible.
Why Are Liquids Used?
Liquids are useful in hydraulic systems because they are nearly incompressible.
If you push on a gas, its particles can be forced closer together and the gas can be compressed considerably.
Liquid particles are already much closer together.
As a result, liquids do not compress easily.
When force is applied to a confined hydraulic liquid, the resulting pressure can therefore be transmitted effectively through the system.
Common hydraulic fluids include specially designed oils.
The Hydraulic System
A basic hydraulic system contains:
- a small input piston
- a large output piston
- a connecting tube
- an enclosed liquid
The small piston has area A₁.
The large piston has area A₂.
A force F₁ is applied to the small piston.
The large piston produces an output force F₂.
Because the pressure is transmitted through the liquid:
P₁ = P₂
Since:
P = F ÷ A
we can write:
F₁ ÷ A₁ = F₂ ÷ A₂
This is the main equation used for simple hydraulic calculations.
How Can Hydraulics Multiply Force?
This is the particularly useful part of Pascal's Principle.
Suppose the input piston is small and the output piston is much larger.
The pressure transmitted through the fluid is the same, but that pressure acts over a much larger area at the output piston.
Because:
Force = Pressure × Area
a larger area produces a larger force.
Therefore:
small input piston → small input force
can produce:
large output piston → large output force
Worked Example 1: A Hydraulic Lift
A hydraulic lift has:
Input piston area = 0.01 m²
Output piston area = 0.20 m²
A force of 100 N is applied to the input piston.
What output force is produced?
Use:
F₁ ÷ A₁ = F₂ ÷ A₂
Substitute:
100 ÷ 0.01 = F₂ ÷ 0.20
First calculate the pressure:
P = 100 ÷ 0.01
P = 10 000 Pa
The same pressure reaches the output piston.
Now:
F₂ = P × A₂
F₂ = 10 000 × 0.20
F₂ = 2000 N
Answer:
Output force = 2000 N
A 100 N input force has produced a 2000 N output force.
The force has been multiplied by:
2000 ÷ 100 = 20
So the system provides a force multiplication of 20 times.
Area Ratio and Force Multiplication
There is a useful shortcut.
The force multiplication depends on the ratio between the piston areas.
For example:
Output area ÷ Input area = 20
means that ideally:
Output force ÷ Input force = 20
Therefore:
20 times larger piston area → 20 times larger force
This gives us an important rule:
The larger the output piston compared with the input piston, the greater the force multiplication.
Worked Example 2: Finding Output Force
A hydraulic machine has:
A₁ = 5 cm²
A₂ = 100 cm²
A force of 40 N is applied to the small piston.
Calculate the output force.
Start with:
F₁ ÷ A₁ = F₂ ÷ A₂
Substitute:
40 ÷ 5 = F₂ ÷ 100
Calculate the pressure ratio:
40 ÷ 5 = 8
Therefore:
F₂ = 8 × 100
F₂ = 800 N
Answer:
Output force = 800 N
Notice that we did not need to convert cm² into m² in this example because both areas used the same units and we were using an area ratio.
Worked Example 3: Finding the Required Input Force
A hydraulic lift needs an output force of 6000 N.
The input piston has an area of 10 cm², while the output piston has an area of 300 cm².
What input force is required?
Use:
F₁ ÷ A₁ = F₂ ÷ A₂
Substitute:
F₁ ÷ 10 = 6000 ÷ 300
6000 ÷ 300 = 20
Therefore:
F₁ = 20 × 10
F₁ = 200 N
Answer:
Only 200 N of input force is needed to produce an ideal output force of 6000 N.
Where Does the Extra Force Come From?
At first, hydraulic force multiplication may appear to create force from nowhere.
A person might push with 200 N and produce an output force of 6000 N.
But hydraulics do not create energy.
There is a trade-off.
The smaller piston must move a greater distance, while the larger piston moves a smaller distance.
For an ideal system:
work input ≈ work output
Since:
Work = Force × Distance
a smaller force acting through a larger distance can produce a larger force acting through a smaller distance.
For example:
small force × large distance
can correspond to:
large force × small distance
So a hydraulic machine trades distance for force.
Conservation of Fluid Volume
There is another way to understand this.
When the small piston moves downward, it pushes a certain volume of liquid through the system.
That same volume must move beneath the larger piston.
The volume moved by a piston is approximately:
Volume = piston area × distance moved
Therefore:
A₁d₁ = A₂d₂
If the output piston has ten times the area, it moves only about one-tenth the distance for the same volume of fluid.
This explains why hydraulic systems can multiply force without creating energy.
Hydraulic Jacks
A hydraulic jack allows a person to lift a heavy vehicle using a relatively small force.
The user applies force to a small piston.
This creates pressure in the hydraulic fluid.
The pressure is transmitted through the fluid to a larger piston.
Because the larger piston has a much greater area, it produces a much larger upward force.
This allows the vehicle to be lifted.
Hydraulic Car Brakes
One of the most important applications of Pascal's Principle is the hydraulic braking system.
When a driver presses the brake pedal:
- The pedal applies force to a piston in the master cylinder.
- The piston increases the pressure of the brake fluid.
- The pressure is transmitted through brake lines.
- The pressure acts on pistons at the wheels.
- These pistons press brake pads against discs or brake shoes against drums.
- Friction slows the wheels.
The hydraulic system allows the driver's input to be transmitted efficiently to multiple wheels.
Why Air in Brake Lines Is a Problem
Hydraulic brake systems are designed to contain liquid rather than air.
Why?
Liquids are difficult to compress.
Gases are much easier to compress.
If air bubbles enter a brake line, some of the driver's input can be used to compress the air instead of effectively transmitting pressure through the system.
The brake pedal may feel soft or spongy, and braking performance can be reduced.
This is why removing air from hydraulic brake lines—called bleeding the brakes—is important.
Hydraulic Lifts
Hydraulic lifts are commonly used in:
- vehicle repair shops
- warehouses
- factories
- elevators
- construction equipment
A relatively small input force can produce enough output force to lift a vehicle or another heavy load.
The principle is the same:
pressure transmitted through fluid + larger output area = larger output force
Hydraulic Excavators
Excavators, loaders, cranes, and other construction machines rely heavily on hydraulics.
Pressurized hydraulic fluid moves pistons inside cylinders.
These cylinders can:
- raise the boom
- move the arm
- operate the bucket
- lift heavy materials
- apply large forces
Hydraulics allow relatively compact machines to generate very large forces.
Hydraulic Presses
A hydraulic press uses Pascal's Principle to produce extremely large forces.
Hydraulic presses are used for:
- shaping metal
- compressing materials
- manufacturing car parts
- forming sheets of metal
- crushing objects
- industrial assembly
The input system creates pressure in hydraulic fluid.
That pressure acts on a large-area piston, creating a very large output force.
Medical Applications
Hydraulic principles can also appear in medical and dental equipment.
Examples include:
- adjustable hospital beds
- patient lifts
- dental chairs
- some surgical equipment
Hydraulic systems are useful because they can produce smooth, controlled movement while supporting large loads.
A Simple Hydraulic Model
A basic hydraulic system can be demonstrated using two syringes connected by flexible tubing and filled with water.
When one syringe is pushed:
- The piston applies force.
- Pressure increases in the water.
- The pressure is transmitted through the tubing.
- The second syringe piston moves.
If syringes of different diameters are used, students can investigate the relationship between piston area, force, and movement distance.
This provides a simple model of real hydraulic machinery.
Comparing the Two Pistons
A typical force-multiplying hydraulic system can be summarized like this:
| Property | Input Piston | Output Piston |
|---|---|---|
| Area | Small | Large |
| Pressure | Same | Same |
| Force | Smaller | Larger |
| Movement distance | Larger | Smaller |
This table highlights an important point:
The pressure is not multiplied.
The force is multiplied because the same pressure acts over a larger area.
A Step-by-Step Method for Hydraulic Problems
When solving a hydraulic problem:
Step 1: Identify the values
Look for:
- F₁
- A₁
- F₂
- A₂
Step 2: Write the relationship
F₁ ÷ A₁ = F₂ ÷ A₂
Step 3: Substitute the known values
Put each value in the correct position.
Step 4: Rearrange if necessary
Solve for the unknown quantity.
Step 5: Check your answer
If the output piston is larger, the output force should normally be larger.
This gives you a quick way to check whether your answer makes sense.
Common Mistakes
Mistake 1: Saying that force is transmitted equally
Pascal's Principle says that a change in pressure is transmitted through a confined fluid.
It does not say that the force must be the same everywhere.
If piston areas are different, the forces can be different.
Mistake 2: Thinking pressure is multiplied
A hydraulic system can multiply force, but in the ideal model the pressure increase is transmitted through the fluid.
P₁ = P₂
The larger force comes from the larger piston area.
Mistake 3: Forgetting the area of the piston
Students sometimes compare piston diameters instead of piston areas.
But the hydraulic equation uses area.
For a circular piston:
A = πr²
A piston with twice the radius has:
4 times the area
not twice the area.
Mistake 4: Thinking hydraulics create energy
A hydraulic system can increase force, but it does not create energy.
The trade-off is movement distance:
larger output force → smaller output movement
Mistake 5: Mixing area units unnecessarily
If one area is given in cm² and another in m², they must be converted to compatible units.
However, if both areas are already in cm², they can often be used directly in:
F₁/A₁ = F₂/A₂
because the area units cancel in the ratio.
Mistake 6: Thinking gases work exactly like hydraulic liquids
Pressure can be transmitted through gases, but gases are easily compressed.
Hydraulic systems generally use liquids because they are much less compressible and provide more predictable force transmission.
Mistake 7: Ignoring friction and energy losses
School calculations usually assume an ideal hydraulic system.
Real systems lose some energy through:
- friction
- fluid resistance
- deformation
- heat
- leakage
Therefore, a real machine may produce slightly less output force than the ideal calculation predicts.
Check Your Understanding
1. Recall
State Pascal's Principle in your own words.
2. Explain
Why are liquids usually used instead of gases in hydraulic systems?
3. Apply
A hydraulic system has a small piston and a large piston.
Why can the large piston produce a greater force even though the pressure is the same?
4. Calculate
A hydraulic system has:
A₁ = 10 cm²
A₂ = 50 cm²
An input force of 60 N is applied.
Calculate the output force.
5. Calculate
A hydraulic lift must produce an output force of 4000 N.
The input piston has an area of 20 cm², and the output piston has an area of 200 cm².
Calculate the required input force.
6. Reason
A hydraulic system multiplies the input force by a factor of 10.
If the small piston moves downward by 20 cm, approximately how far will the large piston move in an ideal system?
7. Application
Explain how Pascal's Principle is used when a driver presses the brake pedal of a car.
8. Challenge
Two circular pistons have radii:
r₁ = 2 cm
r₂ = 6 cm
Without calculating the actual areas, determine how many times larger the area of piston 2 is.
Therefore, approximately how many times larger could the output force be in an ideal hydraulic system?
Key Terms
- Pascal's Principle – a change in pressure applied to a confined fluid is transmitted throughout the fluid
- Hydraulic system – system that uses a confined liquid to transmit pressure and force
- Hydraulic fluid – liquid used to transmit pressure in a hydraulic system
- Input piston – piston where the input force is applied
- Output piston – piston that produces the output force
- Pressure – force acting per unit area
- Force multiplication – production of a larger output force from a smaller input force
- Incompressible – difficult to compress
- Hydraulic cylinder – cylinder containing fluid and a piston used to produce movement and force
- Master cylinder – component that creates hydraulic pressure in systems such as vehicle brakes
Key Takeaways
- Pascal's Principle states that a change in pressure applied to a confined fluid is transmitted throughout the fluid.
- Hydraulic systems usually use liquids because liquids are very difficult to compress.
- In an ideal hydraulic system, P₁ = P₂.
- Hydraulic calculations can use F₁/A₁ = F₂/A₂.
- A larger output piston can produce a larger output force.
- Hydraulic systems multiply force because the same pressure acts over different piston areas.
- Pressure is transmitted; force is multiplied.
- A hydraulic system does not create energy.
- Force multiplication comes with a trade-off: the larger piston moves a smaller distance.
- Pascal's Principle is used in hydraulic brakes, jacks, lifts, presses, excavators, industrial machinery, and medical equipment.