Below is a graph of a displacement-time wave for a particle which is obeying simple harmonic motion.From the graph, estimate the time period, T, the amplitude, A, the frequency, f, and the angular frequency, 𝜔. [4]
2. Sketch the wave resulting from halving the amplitude, and doubling the time period. [2]
The acceleration resulting from simple harmonic motion is given by 𝑎(𝑡)= −𝜔2𝑥(𝑡)
where 𝑥(𝑡) is the displacement as a function of time (the graph above), and 𝜔 is the angular frequency of the wave.
3. Sketch the acceleration-time plot for the graph above. [4]

To analyze the displacement-time wave for a particle undergoing simple harmonic motion (SHM), we can extract several key parameters from the graph. Let's break down the process step by step.

Estimating Key Parameters

From the displacement-time graph, we can determine the following:

Time Period (T)

The time period is the duration it takes for one complete cycle of motion. To find T, look for the distance between two consecutive peaks (or troughs) on the graph. If the distance between these points is, say, 4 seconds, then:

• T = 4 seconds

Amplitude (A)

The amplitude is the maximum displacement from the equilibrium position (the center line of the wave). If the highest point on the graph reaches 3 units above the center line and the lowest point goes to 3 units below, then:

• A = 3 units

Frequency (f)

Frequency is the number of cycles per second and is the inverse of the time period. Using the formula:

• f = 1/T

Substituting our value for T:

• f = 1/4 = 0.25 Hz

Angular Frequency (𝜔)

Angular frequency is related to the frequency by the formula:

• 𝜔 = 2πf

Plugging in the frequency we found:

• 𝜔 = 2π(0.25) ≈ 1.57 rad/s

Modifying the Wave

Next, we need to sketch the wave resulting from halving the amplitude and doubling the time period. If we halve the amplitude from 3 units to 1.5 units, the new wave will peak at 1.5 units above and below the center line. Doubling the time period from 4 seconds to 8 seconds means that the wave will take longer to complete one cycle.

The resulting wave will have a lower peak and a longer duration for each cycle. You would sketch a wave that oscillates between +1.5 and -1.5 units, with each complete cycle taking 8 seconds.

Acceleration-Time Plot

To sketch the acceleration-time plot, we use the formula for acceleration in SHM:

a(t) = -𝜔²x(t)

Since the acceleration is proportional to the negative of the displacement, the acceleration graph will mirror the displacement graph but will be inverted. This means:

• When the displacement is at its maximum (3 units), the acceleration will be at its minimum (negative maximum).
• When the displacement is at zero, the acceleration will also be zero.
• When the displacement is at its minimum (-3 units), the acceleration will be at its maximum (positive maximum).

Thus, the acceleration-time graph will oscillate between the values of -𝜔²(3) and 𝜔²(3), creating a wave that is inverted compared to the displacement wave. The peaks and troughs will occur at the same time intervals as the displacement graph, but the values will be scaled by -𝜔².

In summary, by analyzing the displacement-time graph, we can derive the time period, amplitude, frequency, and angular frequency, and we can also modify the wave parameters and sketch the corresponding acceleration-time plot effectively.