In mechanics, a spring balance measures the weight or force acting on it based on which fundamental law relating the extension of a spring to the applied load?

Introduction / Context:
A spring balance is a simple device commonly used in laboratories and shops to measure weight or force. Inside, it contains a spring that extends when a load is attached. The question asks which fundamental physical law explains how the extension of the spring is related to the applied force. This concept belongs to the study of elasticity and is crucial for understanding how many measuring devices and mechanical systems work.

Given Data / Assumptions:

• The instrument is a spring balance used to measure weight or force.
• The balance contains a coil spring that stretches when a load is applied.
• The question lists several physical laws: Boyle's law, Hooke's law, Bernoulli's principle, and Pascal's law.
• We assume small extensions within the elastic limit of the spring.

Concept / Approach:
Hooke's law is the fundamental law of elasticity for springs. It states that, within the elastic limit, the extension x of a spring is directly proportional to the applied force F, so F = k * x, where k is the spring constant. A spring balance is calibrated based on this relationship so that equal increments of extension correspond to equal increments of force or weight. Boyle's law relates pressure and volume of a gas, Bernoulli's principle deals with energy in flowing fluids, and Pascal's law concerns transmission of pressure in enclosed fluids. None of these gas or fluid laws govern the stretching of a solid spring.

Step-by-Step Solution:
Step 1: Identify the main physical effect in a spring balance: a spring stretches when a weight is attached.Step 2: Recall Hooke's law, which states that extension is proportional to the applied force up to the elastic limit.Step 3: Recognise that the scale markings on a spring balance assume a linear relationship between force and extension.Step 4: Compare this with Boyle's law, Bernoulli's principle, and Pascal's law, which all deal with gases or fluids, not springs.Step 5: Conclude that Hooke's law is the correct principle behind the working of a spring balance.

Verification / Alternative check:
If you hang different known weights on a spring and measure the extension, plotting force on the vertical axis and extension on the horizontal axis gives a straight line passing through the origin for small deformations. This linear graph is experimental evidence of Hooke's law. Spring balances are calibrated in the factory using this relation. If a different law such as Boyle's law or Bernoulli's principle were relevant, you would expect the device to involve gases or flowing fluids, which is not the case. These observations confirm that Hooke's law explains the operation of a spring balance.

Why Other Options Are Wrong:
Boyle's law, option A, describes how pressure and volume of a gas are related at constant temperature and does not involve springs, so it is irrelevant here. Bernoulli's principle, option C, deals with pressure, kinetic, and potential energy in flowing fluids and cannot explain a spring's extension. Pascal's law, option D, is about how pressure is transmitted in confined fluids and applies to hydraulic machines, not to spring balances. Only option B, Hooke's law relating force and extension of a spring, correctly describes the principle used by a spring balance.

Common Pitfalls:
A common mistake is to confuse different physical laws because they are all frequently mentioned in physics. Students might recall Pascal's law or Bernoulli's principle when thinking about pressure or forces without carefully considering the type of system involved. To avoid confusion, always connect the law to its characteristic system: springs with Hooke's law, gases in a cylinder with Boyle's law, fluid flow in pipes and airfoils with Bernoulli's principle, and hydraulic presses with Pascal's law. This mental mapping helps you quickly select the correct principle in instrument based questions.

Final Answer:
Hooke's law relating force and extension of a spring