2.4 Pascal's Principle
Learn how pressure applied to an enclosed fluid is transmitted uniformly in all directions, how hydraulic systems act as force multipliers, and how this principle powers hydraulic jacks and brakes.
Learning Objectives
- State Pascal's principle.
- Explain the transmission of pressure in an enclosed fluid.
- Explain how a hydraulic system acts as a force multiplier.
- Derive and use the formula relating input and output forces and surface areas: \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\).
- Identify applications of Pascal's principle, including hydraulic jacks and hydraulic brake systems.
- Solve numerical problems in daily life involving Pascal's principle.
Basic Hydraulic System as a Force Multiplier
See how a small input force on a narrow piston produces a much larger output force on a wide piston through uniform pressure transmission.
Hydraulic Force Multiplier
Pressure applied at the small input piston is transmitted uniformly through the enclosed hydraulic fluid. The larger output piston area produces a proportionally larger output force.
Key Relationship
Since pressure is transmitted uniformly throughout the enclosed fluid:
Pascal's principle for hydraulic systems
\(\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}\)Multiplying factor \(= \dfrac{A_2}{A_1}\)
When \(A_2 > A_1\), the output force \(F_2\) is greater than the input force \(F_1\). A small input force can therefore lift a heavy load.
Short Explanation
What is Pascal's Principle?
Pascal's principle states that the pressure applied on an enclosed fluid is transmitted uniformly in all directions in the fluid.
Demonstration with Pascal's Piston
When a Pascal's piston filled with water is pushed, water spurts out with equal speed from every hole. This shows that pressure produced on the water surface is transmitted uniformly throughout the water in all directions.
Hydraulic System as a Force Multiplier
A hydraulic system uses a liquid to transmit pressure. Because \(P_1 = P_2\), we get \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\). The output force is \(F_2 = \frac{A_2}{A_1} \times F_1\).
Why Force is Multiplied
Pressure is force per unit area, \(P = \frac{F}{A}\). Keeping the same pressure across a larger output piston area produces a proportionally larger output force. The multiplying factor is \(\frac{A_2}{A_1}\).
| Application | How Pascal's Principle is Used |
|---|---|
| Coconut milk extractor | Uses transmission of pressure in a closed fluid to exert a large force on grated coconut inside a bag to press out coconut milk. |
| Hydraulic jack | Downward force on the handle pushes a small piston. Pressure is transmitted through hydraulic fluid to a large piston, lifting a heavy load. A release valve allows fluid to return to the reservoir when lowering the load. |
| Hydraulic brake system | Pressing the brake pedal exerts force on the master cylinder, creating pressure in the brake fluid. This pressure is transmitted uniformly to slave cylinders at all wheels, pushing brake pads or shoes against the brake disc or drum to stop the vehicle. |
Worked Example 1
\(F_1 = 12\ \mathrm{N},\ A_1 = 10\ \mathrm{cm^2},\ A_2 = 50\ \mathrm{cm^2}\)Multiplying factor \(= \frac{A_2}{A_1} = \frac{50}{10} = 5\).
Output force \(F_2 = 5 \times 12\ \mathrm{N} = 60\ \mathrm{N}\).
Worked Example 2
\(F_1 = 60\ \mathrm{N},\ A_1 = 0.80\ \mathrm{cm^2},\ F_2 = 840\ \mathrm{N}\)\(\frac{840}{A_2} = \frac{60}{0.80}\), so \(A_2 = \frac{840 \times 0.80}{60} = 11.2\ \mathrm{cm^2}\).
Try to Answer First
Answer in your mind, then press “Check Answer”.
What does Pascal's principle state?
Check Answer
In a hydraulic system, an input force of \(4.0\ \mathrm{N}\) acts on a piston with surface area \(0.50\ \mathrm{cm^2}\). Calculate the output force produced on a piston with surface area \(6.4\ \mathrm{cm^2}\).
Check Answer
A small syringe has a piston diameter of \(1.5\ \mathrm{cm}\). To multiply force from \(6\ \mathrm{N}\) to \(72\ \mathrm{N}\), what must be the diameter of the large syringe piston?
Check Answer
Common Mistakes
- Confusing pressure transmission with force transmission: assuming the force is transmitted equally, whereas it is the pressure that is transmitted equally throughout the fluid.
- Forgetting that cross-sectional area relates to diameter squared, \(A = \frac{\pi d^2}{4}\), and directly taking the ratio of diameters instead of the ratio of squared diameters.
- Using inconsistent units between input and output areas, for example mixing \(\mathrm{cm^2}\) and \(\mathrm{m^2}\). Both sides must have matching units.
Concept Misunderstandings
Force is transmitted equally throughout an enclosed liquid.
Pressure is transmitted equally, not force. The output force increases proportionally with the surface area of the output piston.
Pascal's principle works in open containers or gases that leak easily.
Pascal's principle strictly applies to an enclosed fluid where fluid cannot escape and the volume remains confined.
Summary
- Pascal's principle states that pressure applied on an enclosed fluid is transmitted uniformly in all directions.
- For a hydraulic system, \(P_1 = P_2\), so \(\frac{F_1}{A_1} = \frac{F_2}{A_2}\).
- The output force is \(F_2 = \left(\frac{A_2}{A_1}\right) \times F_1\), where \(\frac{A_2}{A_1}\) is the multiplying factor.
- Key applications include the coconut milk extractor, hydraulic jack, and hydraulic brake system.
- A hydraulic jack uses small and large pistons with one-way valves and a fluid reservoir; a hydraulic brake system transmits pressure from the master cylinder to slave cylinders at the wheels.
Short Activity
Objective: Check your understanding of Pascal's principle, hydraulic force multiplication, and hydraulic system components.
A. Objective Quiz
1 Which of the following statements correctly expresses Pascal's principle?
2 In a hydraulic press, an input force of \(15\ \mathrm{N}\) is applied to a piston of cross-sectional area \(3\ \mathrm{cm^2}\). What is the output force exerted on a larger piston with an area of \(30\ \mathrm{cm^2}\)?
B. Fill in the Blanks
3 Pascal's principle states that the pressure applied on an fluid is transmitted uniformly in all directions in the fluid.
4 The multiplying factor of a hydraulic system with an input piston area \(A_1\) and an output piston area \(A_2\) is given by the ratio .
C. Matching / Drag and Drop
Match each component of a hydraulic system or device to its correct function or description. If using a phone, tap the answer first, then tap the matching box.
