Animated Solution for Physics - Oscillations: A particle undergoing simple harmonic motion has time dependent displacement given by $x(t)=A\sin \frac{\pi t}{90}$. The ratio of kinetic to potential energy of this particle at $t=210$ s will be

The problem of finding the ratio of kinetic to potential energy in Simple Harmonic Motion (SHM) is a classic test of both physics intuition and trigonometric agility. Let's break down the journey of this particle and uncover the hidden catch in the options.

We are given a particle executing SHM with a time-dependent displacement:

This equation tells us that the particle starts from the mean position ($x=0$) at $t=0$. The term inside the sine function, $\frac{\pi t}{90}$, represents the phase angle $\omega t$. Our goal is to find the ratio of its kinetic energy ($K$) to its potential energy ($U$) at a specific moment, $t=210$ s.

To find the ratio, we first need the expressions for both energies. The potential energy of a spring-mass system is directly proportional to the square of its displacement:

The kinetic energy, on the other hand, depends on the velocity, which is the derivative of displacement. Thus, it involves the cosine function:

When we take the ratio of kinetic to potential energy, the constants $\frac{1}{2}k{A}^2$ beautifully cancel out, leaving us with a pure trigonometric relation:

This elegant formula is a powerful shortcut for any SHM energy ratio problem!

Now, let's substitute the given time $t=210$ s into our phase angle:

Simplifying the fraction, we get:

To evaluate this easily, we can break it down into a full rotation plus an acute angle:

Since trigonometric functions repeat every $2\pi$, the cotangent of $2\pi +\frac{\pi }{3}$ is exactly the same as the cotangent of $\frac{\pi }{3}$.

We know that $\cot \left(\frac{\pi }{3}\right)=\frac{1}{\sqrt{3}}$. Squaring this value gives us our final answer:

If you look closely at the given options—(a) 2, (b) 1, (c) 1/9, (d) 3—you will notice that 1/3 is missing!

This is a classic example of an error in the question paper. However, notice that option (d) is 3. It is highly probable that the examiner intended to ask for the ratio of potential to kinetic energy ($U/K$), which would indeed be ${\tan }^2(\pi /3)=3$. In competitive exams like JEE, if you are confident in your derivation and spot such a discrepancy, it's best to mark the closest logical intent or leave it for bonus marks.