For a simple pendulum, a graph is plotted between its Kinetic Energy (KE) and Potential Energy (PE)

Q: Why are the graphs parabolas and not straight lines?

Because the restoring force is proportional to the displacement ($F = -kx$). Integrating this force to find work done (which equals potential energy) gives a quadratic relationship ($PE = \frac{1}{2}kx^2$), which graphs as a parabola.

Q: Why does the KE graph open downwards?

Total mechanical energy is constant. Since $KE = E_{total} - PE$, and PE is an upward-opening parabola, subtracting it from a constant gives a downward-opening parabola.

Q: At what displacement are KE and PE equal?

They are equal when $PE = E_{total}/2$. This happens at $d = \pm A/\sqrt{2}$, where $A$ is the maximum amplitude.

Q: What does the horizontal line at the top represent?

It represents the total mechanical energy of the system, which remains constant throughout the oscillation if there is no damping.

Q: Why is option (a) incorrect if it also shows curves?

Option (a) shows curves that are not true parabolas (they are too flat at the bottom). In simple harmonic motion, the energy strictly follows a quadratic (parabolic) path.

Question

For a simple pendulum, a graph is plotted between its Kinetic Energy (KE) and Potential Energy (PE) against its displacement (d) Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

Sigma Percentile · Accepted Answer

### The Energy Dance of a Simple Pendulum

Imagine a simple pendulum swinging back and forth. As it moves, there is a continuous, beautiful exchange between its Kinetic Energy (KE) and Potential Energy (PE). But how exactly do these energies vary with the pendulum's displacement $d$ from its mean position? Let's break down the mathematics behind this energy dance.

### Analyzing the Potential Energy

First, let's look at the Potential Energy. For a simple pendulum undergoing Simple Harmonic Motion (SHM), the restoring force is directly proportional to the displacement, given by $F = -kd$. The potential energy is the work done against this restoring force, which integrates to:

$$PE = \frac{1}{2} k d^2$$

Notice the $d^2$ term? This tells us that the relationship is strictly quadratic. If we plot PE against displacement $d$, we get a perfect **parabola opening upwards**. At the mean position ($d = 0$), the potential energy is at its absolute minimum, which is zero.

### Analyzing the Kinetic Energy

Now, what about the Kinetic Energy? We know from the law of conservation of mechanical energy that the total energy $E_{total}$ of the system remains constant (assuming no air resistance). Therefore, the Kinetic Energy is simply the Total Energy minus the Potential Energy:

$$KE = E_{total} - PE$$

$$KE = \frac{1}{2} k A^2 - \frac{1}{2} k d^2$$

Here, $A$ is the maximum amplitude. Because of the negative sign in front of the $d^2$ term, the graph of KE against displacement is a **parabola opening downwards**. At the mean position ($d = 0$), the pendulum is moving at its fastest, so the Kinetic Energy is at its maximum, equal to the total energy $E_{total}$.

### The Intersection Points

An interesting feature of these graphs is where they intersect. The intersection occurs when Kinetic Energy equals Potential Energy:

$$\frac{1}{2} k d^2 = \frac{1}{2} k (A^2 - d^2)$$

Solving this gives $2d^2 = A^2$, or $d = \pm \frac{A}{\sqrt{2}}$. At these exact points, the energy is split perfectly 50-50 between kinetic and potential forms.

### Final Conclusion

When we compare our derived parabolic curves with the given options, we can confidently eliminate any graphs showing straight lines or non-parabolic curves. Option (b) perfectly illustrates the upward-opening parabola for Potential Energy and the downward-opening parabola for Kinetic Energy, making it the correct representation of the pendulum's energy dynamics.

Animated Solution for Physics - Oscillations: For a simple pendulum, a graph is plotted between its Kinetic Energy (KE) and Potential Energy (PE) against its displacement (d) Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

Select Answer:

Visualized Solution

$E n er g y in S H M$

$P o t e n t ia l Energy Curve$

$K in e t i c Energy Curve$

$I n t er sec t i o n Points$

$C o n c l u s i o n$

$W ha t if plotted against time?$

The Sigma Insight: Force and Energy Method in SHM

The Energy Dance of a Simple Pendulum

Analyzing the Potential Energy

Analyzing the Kinetic Energy

The Intersection Points

Final Conclusion

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