91Ó°ÊÓ

Two blocks \(A\) and \(B\) of mass \(m\) and \(2 m\) respectively are connected by a massless spring of spring constant \(K\). This system lies over a smooth horizontal surface. At \(t=0\) the block. \(A\) has velocity \(u\) towards right as shown while the speed of block \(B\) is zero, and the length of spring is equal its natural length at that instant. $$ \begin{array}{|c|c|} \hline {\begin{array}{c} \text { Column I } \\ \end{array}} & {\begin{array}{c} \text { Column II } \\ \end{array}} \\ \hline \text { i. } \text {The velocity of block A} & \text { a. } \text{can never be zero} \\ \hline \text { ii. } \text{The velocity of block B} & \text { b. } \text{may be zero at certain instants of time} \\ \hline \text { iii. } \text{The kinetic energy of system of two blocks} & \text { c. } \text{is maximum at maximum compression of spring}\\\ \hline \text { iv. } \text{The potential energy of spring} & \text { d. } \text{is maximum at maximum extension of spring} \\ \hline \end{array} $$ Now match the given columns and select the correct option from the codes given below. Codes: $$ \begin{array}{lllll} & \text { i. } & \text { ii. } & \text { iii. } & \text { iv. } \\ \text { (1) } &\text { b }& \text { b} & \text { a, c} & \text { b, d}\\\ \text { (2) } &\text { b }& \text { a, c } & \text { a } & \text { d}\\\ \text { (3) } &\text { d }& \text { a, c } & \text { b } & \text { b, c }\\\ \text { (4) } &\text { a, d }& \text { c } & \text { b } & \text { d } \end{array} $$

Step by step solution

01

Analyze the Velocity of Block A

The initial velocity of block A is given as \( u \) towards the right. As the system only consists of internal forces (spring forces) and there are no external forces acting horizontally, the center of mass velocity remains constant. Hence, the velocity of A changes as the spring compresses/expands, it cannot remain zero for a sustained period because A is lighter and the net force on the two blocks will cause A to continue moving.

02

Analyze the Velocity of Block B

Initially, block B has zero velocity. Since the spring is at its natural length at \( t = 0 \), no net force is acting on B initially. However, as the spring compresses and expands due to the motion induced by block A, block B will acquire some velocity intermittently. Thus, at certain instants, the velocity of block B can return to zero when kinetic energy is exchanged between kinetic and potential forms.

03

Analyze the Kinetic Energy of the System

The kinetic energy of the system is maximum when the spring is neither compressed nor extended. This is due to conservation of energy, where potential energy converts into kinetic energy fully at the point where the spring reaches its natural length again (i.e., zero potential energy). At maximum compression, the potential energy is at its peak, not kinetic energy.

04

Analyze the Potential Energy of the Spring

The spring’s potential energy is maximum when it is at its maximum compression or maximum extension. These are the points where deformation is maximum resulting in maximum potential energy stored within the spring.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spring Constant

In an oscillatory system, the spring constant, represented by the symbol \( K \), is a measure of the stiffness of a spring. This value determines the force required to compress or extend the spring by a unit length along its axis. It is a crucial element in simple harmonic motion (SHM) systems. Understanding the spring constant is vital because it affects the frequency and energy of oscillations in mechanical systems.
Imagine pulling a spring from its natural length and then releasing it. The spring constant gives us an idea of how much force the spring can exert to return to its original form. A large spring constant means the spring is stiffer, needing more force to stretch or compress.

• The relation between force \( F \) and displacement \( x \) is given by Hooke’s Law: \( F = -Kx \).
• This implies that force is directly proportional to displacement in opposite direction.

Kinetic Energy

Kinetic energy in SHM refers to the energy that a body possesses due to its motion. For an object with velocity \( v \) and mass \( m \), the kinetic energy \( KE \) can be expressed as \( KE = \frac{1}{2}mv^2 \). In a system with springs and blocks, the kinetic energy varies as the system oscillates.
During the motion, the total mechanical energy is conserved. This means that as blocks move, any decrease in kinetic energy leads to an equivalent increase in potential energy, and vice versa. However, the kinetic energy is maximum when the spring is at its natural length.

• Maximum kinetic energy occurs when the potential energy is minimum.
• The system's motion is influenced by the conservation of energy principles.

Potential Energy

Potential energy in SHM is stored due to the position of an object within a force field, such as a spring. For a spring, the potential energy \( PE \) at any point is given by \( PE = \frac{1}{2}Kx^2 \), where \( x \) is the displacement of the spring from its natural length.
As a spring compresses or extends, it stores energy that can be converted back to kinetic form. This energy reaches its maximum either at the points of maximum compression or extension because that's when the deformation is greatest.

• Potential energy is highest when the spring is stretched or compressed to the extreme positions.
• It converts completely into kinetic energy when the spring returns to its equilibrium position.

Mass and Velocity Relation

The relation between mass and velocity in a system in SHM is pivotal in understanding the dynamics of the system. When two objects, like block A and B in the exercise, are connected by a spring, their motion and speed are interconnected due to conservation of momentum.
In our example, block A starts with an initial velocity \( u \) while block B does not move initially. When the spring is set in motion, energy exchanges between the two blocks due to changes in the spring's length.

• Block A and B’s velocities are related such that the center of mass of the system remains constant.
• Block A being lighter, tends to move more quickly, while Block B being heavier can slow down or stop momentarily when velocities are exchanged between potential and kinetic energy forms.