Question

Assertion and Reason are given in following questions. Each question have four option. One of them is correct it. (1) If both assertion and reason and the reason is the correct explanation of the Assertion. (2) If both assertion and reason are true but reason is not the correct explanation of the assertion. (3) If the assertion is true but reason is false. (4) If the assertion and reason both are false. Assertion: Both, a stretched spring and a compressed spring have potential energy. Reason: Work is done against the restoring force in each case. (A) 1 (B) 2 (C) 3 (D) 4

Short answer

The correct answer is (A) 1, as both the Assertion and the Reason are true, and the Reason is the correct explanation for the Assertion.

Step-by-step solution

Understanding the Assertion

: A stretched spring and a compressed spring both have potential energy because they have the capacity to do work when allowed to return to their equilibrium positions. The potential energy stored in the spring is given by the formula: \(PE = \frac{1}{2}kx^2\), where \(PE\) is the potential energy, \(k\) is the spring constant, and \(x\) is the displacement from the equilibrium position. Since the formula does not differentiate between stretched or compressed springs (positive or negative displacement from equilibrium), we can conclude that the Assertion is true.

Understanding the Reason

: The restoring force that a spring exerts is proportional to its displacement from the equilibrium position and acts in the opposite direction of the displacement (according to Hooke's Law: \(F = -kx\)). When we stretch or compress a spring, we perform work against this restoring force to increase the spring's potential energy. This means that we are transferring energy from our muscles to the spring in the form of potential energy, and the spring can later release this energy when returning to its equilibrium position. Thus, we can conclude that the Reason is also true.

Comparing the Assertion and Reason

: Now that we have established the truth of both the Assertion and the Reason, we can evaluate whether the Reason is indeed the correct explanation for the Assertion. The Assertion states that both a stretched and a compressed spring have potential energy, and the Reason provides an explanation involving the work done against the restoring force when stretching or compressing a spring. This explanation is consistent with the Assertion and directly supports it. Therefore, the Reason is the correct explanation for the Assertion. Since both the Assertion and the Reason are true, and the Reason is the correct explanation for the Assertion, the correct answer is (A) 1.

Key concepts

Potential Energy

Potential energy is the stored energy in an object due to its position or state. For springs, this energy becomes relevant when the spring is either stretched or compressed from its natural, or equilibrium, length. This energy is not active but can be transformed into kinetic energy if the spring is allowed to move back to its equilibrium position. The potential energy in a spring can be calculated using the formula \[ PE = \frac{1}{2}kx^2 \] where:

• \(PE\) stands for potential energy
• \(k\) is the spring constant
• \(x\) represents the displacement from equilibrium

This formula indicates how even a small displacement can store a significant amount of energy due to the square of the displacement, making understanding potential energy crucial for grasping how springs operate and how they store energy.

Hooke's Law

Hooke's Law is fundamental in understanding springs and their behavior. It states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. This relationship is mathematically represented as: \[ F = -kx \] where:

• \(F\) is the restoring force
• \(k\) is the spring constant
• \(x\) is the displacement from the equilibrium position

The negative sign indicates that the force exerted by the spring is opposite in direction to the displacement. This means if you stretch the spring, it will pull back, and if you compress it, it will push out. The spring constant \(k\) is crucial as it determines the stiffness of the spring. A higher value of \(k\) means a stiffer spring requiring more force for the same displacement.

Restoring Force

The restoring force is the force exerted by a spring that aims to bring it back to its equilibrium position. This force is what makes springs useful in applications where elasticity and energy storage are essential. When a spring is displaced—either stretched or compressed—it creates this restoring force that acts to return the spring to its original state. The magnitude of the restoring force is significant because:

• It is directly proportional to how far the spring has been displaced, as described by Hooke's Law (\(F = -kx\)).
• It is a vector quantity, meaning it has both magnitude and direction, with the direction being opposite to the displacement.

Understanding the restoring force helps explain how energy is stored in and released from a spring during its use.

Work Done

Work done on or by a spring involves moving the spring from its equilibrium position by stretching or compressing it. The work done can be thought of as the effort you input to change the spring's state, which results in the spring storing energy as potential energy. The formula for work done in spring deformation is directly linked to the potential energy: \[ W = \frac{1}{2}kx^2 \] where:

• \(W\) is the work done
• \(k\) is the spring constant
• \(x\) is the deformation from the equilibrium position

The work done on the spring can either be positive, meaning energy is added to the system by stretching or compressing the spring, or negative, when the spring does work by returning to its original shape and transferring energy back.

Spring Constant

The spring constant \(k\) is a measure of a spring's stiffness or rigidity. It represents how resistant a spring is to being deformed by an applied force. The value of the spring constant directly affects how much force is needed for a certain amount of stretch or compression according to Hooke's Law: \[ F = -kx \]where:

• \(F\) is the force applied
• \(k\) is the spring constant
• \(x\) is the displacement from its equilibrium position

A larger \(k\) value means the spring is stiffer, requiring more force to achieve the same amount of stretch or compression compared to a spring with a lower \(k\). Understanding the spring constant is crucial for designing systems that require specific stiffness or flexibility, ensuring the correct response under various forces.