In automobile engineering, a hydraulic brake system works based on which fundamental law of fluid pressure?

Introduction / Context:
Hydraulic brake systems are widely used in cars, motorcycles and other vehicles. When the driver presses the brake pedal, a small force applied at one point is transmitted through brake fluid to generate a large braking force at the wheels. This question tests your understanding of which physical law explains how pressure is transmitted undiminished in an enclosed fluid, allowing force multiplication in hydraulic devices such as brakes, jacks and lifts.

Given Data / Assumptions:
- The system mentioned is a hydraulic brake, which uses an incompressible fluid in a closed system.
- The options include Pascal's law, Boyle's law, Coulomb's law and Bernoulli's law.
- We assume normal conditions and ideal behaviour of the brake fluid with negligible compressibility and no leaks.

Concept / Approach:
Pascal's law states that any change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel. This principle allows a small input force on a small-area piston to produce a large output force on a larger-area piston. Boyle's law relates pressure and volume of a gas at constant temperature. Coulomb's law deals with electric force between charges, and Bernoulli's law describes energy conservation in flowing fluids. Only Pascal's law directly describes how hydraulic brakes operate.

Step-by-Step Solution:
Step 1: Identify the working medium in hydraulic brakes. It is a liquid (brake fluid) in a closed system. Step 2: Recall Pascal's law: pressure applied at one point in an enclosed fluid is transmitted equally in all directions throughout the fluid. Step 3: When the driver presses the brake pedal, a force is applied to a small master cylinder, creating pressure in the fluid. Step 4: This pressure is transmitted through brake lines to larger slave cylinders at the wheels, multiplying the force and pressing brake pads against the discs or drums. Step 5: Since this behaviour matches Pascal's law, that law is the correct answer.

Verification / Alternative check:
Consider a simple hydraulic press with two pistons of areas A1 and A2. If a force F1 on the small piston creates pressure P = F1 / A1, Pascal's law states that the same pressure acts on the larger piston, giving F2 = P * A2 = (F1 / A1) * A2. This shows how small input forces become large output forces, exactly what happens in hydraulic brakes. Boyle's law, Coulomb's law and Bernoulli's law do not explain this type of static fluid pressure transmission in a closed system.

Why Other Options Are Wrong:
Boyle's law: Deals with pressure and volume of a gas at constant temperature, not with transmission of pressure in a liquid and force multiplication.
Coulomb's law: Describes electrostatic forces between charges and has nothing to do with hydraulic systems or pressure transmission in fluids.
Bernoulli's law: Relates pressure, kinetic energy and potential energy in a moving fluid, mainly for flow situations like in pipes and airfoils, not for static hydraulic brake systems.

Common Pitfalls:
Students sometimes confuse Bernoulli's law and Pascal's law because both involve fluids. The key difference is that Pascal's law deals with static enclosed fluids and equal transmission of pressure, while Bernoulli's law applies to moving fluids and conservation of energy along a streamline. Remembering that brakes are essentially static pressure devices helps you associate them correctly with Pascal's law.

Final Answer:
A hydraulic brake works on the principle of Pascal's law.