Animated Solution for Physics - Oscillations: A block of mass $1\text{ kg}$ attached to a spring is made to oscillate with an initial amplitude of $12\text{ cm}$. After $2\text{ min}$, the amplitude decreases to $6\text{ cm}$. Determine the value of the damping constant for this motion. (Take, $\ln 2=0.693$)

Imagine a block attached to a spring, oscillating back and forth. In an ideal, frictionless world, this block would continue to oscillate forever with the same amplitude. However, in reality, forces like air resistance or internal friction within the spring gradually drain energy from the system. This phenomenon is known as damped oscillation.

As the energy dissipates, the maximum displacement of the block—its amplitude—shrinks over time. This decay isn't linear; it follows a beautiful exponential curve. In our specific problem, we observe a block starting with an initial amplitude of $12\text{ cm}$, which gracefully decays to exactly half its value, $6\text{ cm}$, over a span of $2\text{ minutes}$.

To uncover the physical properties of this damping, we rely on the master equation for the amplitude of a damped harmonic oscillator. The amplitude $A$ at any given time $t$ is expressed as:

Here, $A_0$ represents the initial amplitude, $m$ is the mass of the oscillating block, and $b$ is the damping constant—the exact parameter we are tasked to find. The damping constant quantifies how strongly the surrounding medium resists the motion.

Let's carefully substitute the physical realities of our problem into this equation. We know the final amplitude $A=6\text{ cm}$ and the initial amplitude $A_0=12\text{ cm}$. The mass $m$ is simply $1\text{ kg}$.

Crucially, we must ensure dimensional consistency. The time given is $2\text{ minutes}$, which we must convert to SI units. Therefore, $t=120\text{ seconds}$. Substituting these values yields:

Now, we embark on the algebraic manipulation to isolate $b$. First, we divide both sides of the equation by $12$:

This simplifies neatly to:

To make the equation even cleaner and avoid dealing with negative exponents, we can take the reciprocal of both sides:

To bring the unknown variable $b$ down from the exponent, we apply the natural logarithm ($\ln$) to both sides of the equation. The natural logarithm is the inverse operation of the exponential function $e^x$.

The problem kindly provides the value of $\ln 2$ as $0.693$. With this, finding $b$ is just a matter of simple division:

Converting this into scientific notation to match our options, we get:

This is our damping constant! It's fascinating to note that the time it takes for the amplitude to halve is a constant for a given system, much like the half-life of a radioactive isotope. If we were to increase this damping constant $b$—perhaps by immersing the block in a thicker fluid like oil—the oscillations would die out much more rapidly.