Animated Solution for Physics - Oscillations: The displacement of a damped harmonic oscillator is given by $x(t)=e^{-0.1t}\cos (10\pi t+\varphi )$. Here, $t$ is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to

Imagine a pendulum swinging in a jar of honey. Unlike a pendulum in a vacuum that swings forever, the honey exerts a drag force, slowly sapping the pendulum's energy. This is the essence of a damped harmonic oscillator.

The mathematical signature of such a system is a displacement equation that combines two distinct behaviors: oscillation and decay. In our problem, the displacement is given by:

This equation is a beautiful marriage of two functions. The $\cos (10\pi t+\varphi )$ part represents the back-and-forth oscillation. If it were alone, the amplitude would always be $1$. However, it is multiplied by the term $e^{-0.1t}$. This exponential term acts as an "envelope," squeezing the cosine wave tighter and tighter as time goes on.

To find out how the maximum swing (the amplitude) changes over time, we can completely ignore the rapidly fluctuating cosine term. The amplitude at any given time $t$ is dictated entirely by the exponential coefficient:

Our mission is to find the exact moment when this amplitude drops to half of its starting value. But what is the starting value? We simply plug in $t=0$:

So, the initial amplitude is exactly $1$ unit.

We want to find the time $t$ when the amplitude $A(t)$ becomes half of $A_0$, which is $1/2$. We set up our equation:

Dealing with negative exponents can sometimes lead to silly algebraic mistakes. A neat trick is to take the reciprocal of both sides. The reciprocal of $e^{-0.1t}$ is $e^{0.1t}$, and the reciprocal of $1/2$ is $2$. This gives us a much friendlier equation:

To rescue our variable $t$ from the exponent, we apply the natural logarithm ($\ln$) to both sides. The natural log and the exponential function $e$ are mathematical inverses, so they cancel each other out on the left side:

Now, it's just a matter of simple arithmetic. We isolate $t$ by dividing by $0.1$ (which is the same as multiplying by $10$):

In physics and chemistry, the value of $\ln (2)$ pops up everywhere, from radioactive half-lives to first-order chemical kinetics. It is highly recommended to memorize its approximate value: $\ln (2)\approx 0.693$.

Substituting this value in, we get:

Looking at our multiple-choice options, the closest value to $6.93$ seconds is $7\text{ s}$. The amplitude of our damped oscillator will be halved in approximately $7$ seconds.