Animated Solution for Physics - Oscillations: Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II.

In this problem, we are presented with a classic JEE Advanced Matrix Match question that tests our fundamental understanding across multiple domains of physics: Simple Harmonic Motion (SHM), differential equations of motion, pseudo forces in non-inertial frames, and gravitational escape velocity.

Let's break down each scenario in Column I systematically and match them with the correct characteristics in Column II.

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The Mathematical Signature of SHM

We are given that the object moves on the $x$-axis under a conservative force such that its speed and position satisfy:

where $c_1$ and $c_2$ are positive constants.

To understand the nature of this motion, let's square both sides of the equation:

Now, let's recall the standard velocity-displacement relation for a particle executing Simple Harmonic Motion (SHM) with angular frequency $\omega$ and amplitude $A$:

By direct comparison, we can map the constants:

Since the velocity equation of our object is mathematically identical to that of a simple harmonic oscillator, the object must execute Simple Harmonic Motion.

Thus, (A) matches with (p).

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Exponential Decay and Monotonic Motion

In the second scenario, the object's velocity and displacement satisfy:

where $k$ is a positive constant. Since velocity is the rate of change of position, we can write this as a first-order differential equation:

Separating the variables to solve this differential equation:

Integrating both sides from $t=0$ (where $x=x_0$) to any time $t$:

Taking the exponential of both sides gives the position as a function of time:

Now, let's find the velocity as a function of time:

Let's analyze the physical consequences of these equations:

1. Direction of Motion: Since the exponential term $e^{-kt}$ is strictly positive for all finite values of $t$, the velocity $v(t)$ never changes its sign. This means the object continues to move in its initial direction indefinitely and does not change its direction.

2. Kinetic Energy: The kinetic energy $K$ of the object is given by:

As time $t$ increases, the term $e^{-2kt}$ decreases monotonically towards zero. Therefore, the kinetic energy of the object keeps on decreasing.

Thus, (B) matches with (q) and (r).

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Spring-Mass System in an Accelerating Frame

Here, a spring-mass system is observed from inside an elevator accelerating upwards with a constant acceleration $a$.

Let's analyze the forces acting on the mass $m$ in the non-inertial frame of the elevator:

1. Gravity: A downward gravitational force $mg$. 2. Pseudo Force: Since the elevator is accelerating upwards, a constant pseudo force $ma$ acts downwards on the mass. 3. Spring Force: A restoring force $-kx$ acting towards the unstretched position of the spring.

The effective downward force on the mass at any displacement $x$ from the unstretched position is:

At the new equilibrium position (mean position), the net force is zero:

If we displace the block by a small distance $y$ from this new equilibrium position ($x=x_0+y$), the net restoring force becomes:

This is the classic restoring force equation for Simple Harmonic Motion!

Key Takeaway: A constant external force (like gravity or a constant pseudo force) only shifts the mean position of oscillation but never alters the simple harmonic nature or the time period of the system.

Thus, (C) matches with (p).

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High-Speed Projectile Escaping Gravity

In the final scenario, a projectile is launched vertically upwards from the Earth's surface with a speed:

Let's recall the expression for the escape velocity $v_e$ from the surface of the Earth:

Comparing our launch speed $v$ with the escape velocity:

Since the projection speed is strictly greater than the escape velocity ($v>v_e$), the projectile has more than enough kinetic energy to overcome the Earth's gravitational pull completely and escape to infinity.

Let's analyze its motion:

1. Direction of Motion: Since it escapes to infinity, it will keep moving upwards forever. It will never stop, reverse, or turn back. Thus, the object does not change its direction.

2. Kinetic Energy: As the projectile moves away from the Earth, the gravitational force continuously pulls it downwards (opposite to its velocity). This gravitational force does negative work on the projectile, continuously reducing its speed and kinetic energy all the way to infinity.

Thus, (D) matches with (q) and (r).