The displacement-time graph of a particle executing SHM is given in figure (sketch is schematic and

Q: Why does the graph start at a maximum instead of zero?

The starting position depends on the initial phase of the particle. If a particle is released from the extreme position at $t=0$, its displacement is maximum, which corresponds to a cosine function $x(t) = A \cos(\omega t)$. If it started from the mean position, it would be a sine function.

Q: How do we know the force is zero when the displacement is zero?

According to Hooke's Law for simple harmonic motion, the restoring force is directly proportional to the displacement ($F = -kx$). When the particle passes through the mean position ($x=0$), the spring is neither compressed nor stretched, so the force is exactly zero.

Q: Isn't acceleration zero when the particle is at the extreme position because it stops?

This is a very common misconception! While the velocity is momentarily zero at the extreme position, the acceleration is actually at its maximum. Acceleration depends on the force ($a = F/m$), and the restoring force is strongest when the displacement is maximum.

Q: Why is the speed maximum at the mean position?

By the conservation of mechanical energy, the total energy of the system is constant. At the mean position, the potential energy is zero (since $x=0$). Therefore, all the total energy is converted into kinetic energy, making the speed maximum.

Q: When are the kinetic and potential energies actually equal?

They are equal when $\frac{1}{2}kx^2 = \frac{1}{2}k(A^2 - x^2)$, which simplifies to $x = \pm \frac{A}{\sqrt{2}}$. In terms of time for a cosine wave, this occurs at $t = \frac{T}{8}, \frac{3T}{8}, \frac{5T}{8}$, and so on, not at the extreme positions.

Question

The displacement-time graph of a particle executing SHM is given in figure (sketch is schematic and not to scale)

Which of the following statement(s) is/are true for this motion?
A. The force is zero at $t = \frac{3T}{4}$.
B. The acceleration is maximum at $t = T$.
C. The speed is maximum at $t = \frac{T}{4}$.
D. The potential energy is equal to kinetic energy of the oscillation at $t = \frac{T}{2}$.

Options:

(A) A, B and D

(B) B, C and D

(C) A, B and C

(D) A and D

Sigma Percentile · Accepted Answer

### Decoding the Displacement-Time Graph

When tackling graphical problems in Simple Harmonic Motion (SHM), the first step is to carefully observe the initial conditions. Looking at the given displacement-time graph, we can see that at time $t = 0$, the particle is at its maximum positive displacement ($+A$). This immediately tells us that the motion can be described by a cosine function:

$$x(t) = A \cos(\omega t)$$

With this foundational understanding, we can evaluate each statement by locating the specific time on the graph and analyzing the physical state of the particle.

### Analyzing Statement A: Force at the Mean Position

Statement A asks us to evaluate the force at $t = \frac{3T}{4}$. If we trace this time on the horizontal axis, we see that the graph intersects the time axis. This means the displacement is exactly zero ($x = 0$), placing the particle at the **mean position**.

In SHM, the restoring force is governed by Hooke's Law:

$$F = -kx$$

Since the displacement $x$ is zero, the restoring force $F$ must also be zero. Therefore, Statement A is absolutely correct.

### Analyzing Statement B: Acceleration at the Extreme

Next, let's look at Statement B, which claims the acceleration is maximum at $t = T$. Checking the graph at $t = T$, we find the particle at the peak of the curve, which is the **positive extreme position** ($x = +A$).

The acceleration of a particle in SHM is given by:

$$a = -\omega^2 x$$

At the extreme position, the magnitude of displacement is at its maximum ($A$). Consequently, the magnitude of acceleration is also at its maximum ($|a| = \omega^2 A$). The spring (or restoring mechanism) is stretched to its absolute limit, providing the maximum pull back towards the center. Thus, Statement B is correct.

### Analyzing Statement C: Speed at the Mean Position

Statement C discusses the speed at $t = \frac{T}{4}$. Looking at the graph, the curve crosses the time axis at this instant, meaning the particle is once again at the **mean position** ($x = 0$).

The velocity of a particle in SHM is related to its displacement by the equation:

$$v = \omega \sqrt{A^2 - x^2}$$

When $x = 0$, the expression simplifies to $v = \omega A$, which is the maximum possible speed. Physically, as the particle passes through the equilibrium point, all of its potential energy has been converted into kinetic energy. Therefore, Statement C is correct.

### Analyzing Statement D: The Energy Balance

Finally, Statement D asserts that the potential energy equals the kinetic energy at $t = \frac{T}{2}$. At $t = \frac{T}{2}$, the graph shows the particle at the **negative extreme position** ($x = -A$).

At any extreme position, the particle momentarily comes to a halt before reversing its direction. Because its velocity is zero, its **kinetic energy is zero** ($KE = 0$). Conversely, the spring is fully compressed, meaning the **potential energy is at its maximum** ($PE = \frac{1}{2}kA^2$).

Clearly, $0 
eq \frac{1}{2}kA^2$, so the kinetic and potential energies are not equal at this time. Statement D is incorrect.

### The Final Verdict

After a thorough graphical analysis, we have determined that Statements A, B, and C are true, while Statement D is false. This leads us directly to the correct option, which is **(c)**.

Select Answer:

Visualized Solution

$x - t$ Graph Analysis

Statement A: Position at $t = \frac{3 T}{4}$

Statement A: Force Evaluation

Statement B: Position at $t = T$

Statement B: Acceleration Evaluation

Statement C: Position at $t = \frac{T}{4}$

Statement C: Speed Evaluation

Statement D: Position at $t = \frac{T}{2}$

Statement D: Energy Evaluation

Final Conclusion

The Way Forward

The Sigma Insight: Simple Harmonic Motion (SHM)

Decoding the Displacement-Time Graph

Analyzing Statement A

Analyzing Statement B

Analyzing Statement C

Analyzing Statement D

The Final Verdict

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