Problem 21 - Master Resource Book in JEE Main Physics

Question

A spring with one end attached to a mass and the other to a rigid support is stretched and released [NCERT Exemplar] (a) Magnitude of acceleration, when just released is maximum (b) Magnitude of acceleration, when at equilibrium position is maximum (c) Speed is maximum when mass is at equilibrium position (d) Magnitude of displacement is always maximum whenever speed is minimum

Step-by-Step Solution

Answer

(a) True, (b) False, (c) True, (d) True.

Step 1: Understanding the Problem

We have a system where a spring is stretched and released, with a mass attached to it. We need to identify statements about the motion such as acceleration, speed, and displacement when related to equilibrium position.

Step 2: Reviewing Hooke’s Law and Newton's Second Law

Recall that the force exerted by a spring is given by Hooke’s law: \(F = -kx\), where \(k\) is the spring constant and \(x\) is the displacement from the equilibrium position. According to Newton's second law, \(F = ma\), where \(m\) is the mass and \(a\) is the acceleration.

Step 3: Determining When Acceleration is Maximum

Based on the formula \(a = \frac{-kx}{m}\), acceleration is maximum when \(x\) (displacement from the equilibrium) is maximum. This happens when the mass is at the stretched position, just released.

Step 4: Identifying Speed at Equilibrium Position

Speed is related to kinetic energy, which is maximum when potential energy is minimum. At the equilibrium position, all the energy is kinetic (speed is maximum) since displacement \(x = 0\) and potential energy is zero.

Step 5: Connecting Displacement and Speed

Displacement is maximum when the speed of the mass is zero, such as when the spring is either fully compressed or stretched. This is when the system's energy is entirely potential.

Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in classical mechanics that describes the linear relationship between the force exerted by a spring and the displacement from its equilibrium position. Simply put, the more you stretch or compress a spring, the greater the force it exerts. Mathematically, it's expressed as:

• \( F = -kx \)

Here:

• \( F \) is the force exerted by the spring,
• \( k \) is the spring constant (a measure of the spring's stiffness), and
• \( x \) is the displacement from the equilibrium position.

Negative sign indicates that the spring force is a restoring force, meaning it always acts in the direction opposite to the displacement. This law only holds up to the elastic limit of the spring, beyond which permanent deformation occurs.
Understanding Hooke's Law helps predict how springs behave in simple harmonic motion, making it a core topic in both physics and engineering.

Newton's Second Law

Newton's Second Law of Motion forms a cornerstone of classical mechanics and can be concisely expressed by the equation:

• \( F = ma \)

This equation denotes that the force acting upon an object is equal to the mass (\( m \)) of the object multiplied by its acceleration (\( a \)). Essentially, it describes how the velocity of an object changes when it is subjected to an external force.
In the context of simple harmonic motion, it combines beautifully with Hooke’s Law to explain the acceleration of a mass attached to a spring:

• \( a = \frac{-kx}{m} \)

This means:

• Acceleration is directly proportional to displacement \( (x) \) and inversely proportional to the mass of the object.
• The direction of the acceleration is opposite to the displacement, approving the restoring nature of the spring force.

Understanding this law allows us to grasp why acceleration reaches its peak when a spring is at its maximum stretch or compression. It underscores how force causes objects to accelerate or change their motion.

Equilibrium Position

The equilibrium position is a crucial concept in the study of oscillations and waves. It refers to the point where the forces on a system are balanced, and in the absence of disturbances, the system would remain at rest. For a mass-spring system:

• This is where the net force is zero, as the spring is neither stretched nor compressed.
• The displacement from this position is zero \( (x = 0) \).

When the mass is at this point in its oscillatory motion, potential energy is at its minimum, and kinetic energy, and thus speed, are at their maximum. This balance of forces and continuous transformation of energy between kinetic and potential forms, makes the equilibrium position a dynamic focal point in simple harmonic motion.
By analyzing this position, we can better predict the behavior of oscillating bodies and understand how energy transfer occurs in such systems.

Acceleration and Displacement

In simple harmonic motion, acceleration and displacement are intimately linked. The displacement refers to how far the mass is from the equilibrium position, and acceleration is how quickly the speed of the mass changes.

• Acceleration is maximum when the displacement is at its peak (either fully compressed or stretched), meaning it's most intense when the spring is most out of shape.
• As per the formula \( a = \frac{-kx}{m} \), acceleration is dependent on both the displacement \( (x) \) and the spring constant \( (k) \).

This relationship ensures that the acceleration is always directed towards the equilibrium position, serving as a restoring force that tries to bring the mass back to the center. The periodic nature of this relationship defines the characteristic oscillatory motion seen in systems demonstrating simple harmonic motion (SHM).
Understanding this linkage helps in analyzing how quickly a system can respond to forces, and in designing systems to withstand certain oscillatory conditions.

Kinetic and Potential Energy

In a mass-spring system undergoing simple harmonic motion, energy constantly shifts between kinetic and potential forms. This interchange is crucial in understanding the motion dynamics:
**Kinetic Energy (KE)**:

• It is maximum at the equilibrium position where velocity is highest.
• Expressed as \( KE = \frac{1}{2} mv^2 \), it depends on the mass \( (m) \) and velocity \( (v) \) of the mass.

**Potential Energy (PE)**:

• It is maximum when the spring is either fully compressed or stretched.
• Given by \( PE = \frac{1}{2} kx^2 \), it is reliant on the spring constant \( (k) \) and the displacement \( (x) \) from the equilibrium.

The total mechanical energy in the system remains constant, although its form changes back and forth between kinetic and potential. This conservation principle is a powerful concept in physics, revealing how energy is neither created nor destroyed but transformed.
Grasping the ebb and flow of kinetic and potential energy aids in predicting the speed and position of the mass at any given time.