Animated Solution for Physics - Oscillations: A particle is executing simple harmonic motion (SHM) of amplitude $A$, along the X-axis, about $x=0$. when its potential energy (PE) equals kinetic energy (KE), the position of the particle will be

Imagine a child on a swing, soaring back and forth. At the highest point of the swing, for a brief, magical fraction of a second, the child is completely still. All the energy of motion has been converted into the potential to fall back down. As the swing rushes through the lowest point, it moves at its absolute fastest—all that potential energy has transformed into kinetic energy.

This continuous, elegant exchange between Potential Energy (PE) and Kinetic Energy (KE) is the defining characteristic of Simple Harmonic Motion (SHM). But a fascinating question arises: at what exact point in this journey are the two energies perfectly balanced? Where does $\text{PE}=\text{KE}$?

To find this point, we must first understand how these energies depend on the particle's position, $x$.

The Potential Energy in a spring-mass system (or any SHM) is given by the formula: This equation tells us that PE is a parabola that opens upwards. It is zero at the mean position ($x=0$) and reaches its maximum at the extreme positions ($x=\pm A$).

On the other hand, the total mechanical energy $E$ of the system is constant and is equal to the maximum potential energy:

Since energy is conserved, the Kinetic Energy ($K$) at any point is simply the total energy minus the potential energy: Graphically, this is an inverted parabola. It peaks at the center and drops to zero at the extremes.

We are looking for the specific position $x$ where the potential energy is exactly equal to the kinetic energy. Mathematically, we just set our two expressions equal to each other:

This equation might look a bit dense, but it simplifies beautifully. Notice that the term $\frac{1}{2}k$ appears on both sides. We can divide the entire equation by $\frac{1}{2}k$, effectively canceling it out:

Now, let's group the $x^2$ terms together by adding $x^2$ to both sides:

Dividing by 2, we isolate $x^2$:

Finally, taking the square root of both sides gives us our answer:

What does this result mean? First, the $\pm$ sign tells us there are two such points—one on the right side of the mean position and one on the left. This makes perfect sense because SHM is completely symmetric.

Second, notice that the answer is not $x=\frac{A}{2}$. A common trap is to assume that halfway in energy means halfway in distance. But because energy scales with the square of the distance, the point of equal energy is pushed further out. Since $\sqrt{2}\approx 1.414$, the position is roughly $x\approx \pm 0.707A$. The particle has to travel about 70.7% of the way to the extreme before its potential energy catches up to its kinetic energy!