Animated Solution for Physics - Oscillations: A body executes simple harmonic motion. The potential energy (PE), the kinetic energy (KE) and total energy (TE) are measured as function of displacement $x$. Which of the following statements is true?

Imagine you are sitting on a swing. As you pump your legs and soar through the air, you are experiencing one of the most beautiful phenomena in physics: Simple Harmonic Motion (SHM). But what is really happening behind the scenes? It is a continuous, elegant dance between two forms of energy—Kinetic and Potential.

In any SHM system, whether it's a pendulum, a mass on a spring, or a swing, the total mechanical energy remains absolutely constant. This is the law of conservation of energy. However, the distribution of this energy changes depending on where the object is in its path.

Let's look at the mathematical heartbeat of this system. If a particle of mass $m$ is oscillating with an angular frequency $\omega$ and an amplitude $a$, its displacement from the center (the mean position) is given by $x$.

The Potential Energy (PE), which is the energy stored due to the object's position, is given by:

The Kinetic Energy (KE), which is the energy of motion, depends on the object's velocity. Expressed in terms of displacement, it is:

Notice how these two equations are perfectly complementary. If you add them together, the $x^2$ terms cancel out, leaving the Total Energy (TE):

This Total Energy is a constant value, completely independent of $x$.

Now, let's focus on the exact center of the oscillation—the mean position. Here, the displacement $x=0$.

What happens to our energies at this specific point? Let's substitute $x=0$ into our Potential Energy equation:

The Potential Energy drops to zero. All the stored energy has been released. But where did it go? It transformed entirely into Kinetic Energy. Let's verify this by substituting $x=0$ into the Kinetic Energy equation:

Look at that! The Kinetic Energy is now equal to the Total Energy. Because the term being subtracted ($-x^2$) is at its absolute minimum (zero), the Kinetic Energy is at its absolute maximum.

Does this make physical sense? Absolutely. Think back to the swing. When are you moving the fastest? Right at the bottom of the arc, as you rush past the center point. At this mean position, your velocity is maximum, and therefore, your Kinetic Energy is maximum.

Conversely, when you reach the highest point of your swing (the extreme position where $x=a$), you momentarily stop. Your velocity is zero, meaning your Kinetic Energy is zero. At that exact moment, all your energy is stored as Potential Energy, ready to pull you back down.

Therefore, the statement "KE is maximum when $x=0$" is the correct physical and mathematical reality of Simple Harmonic Motion.