To determine which graph correctly plots the displacement $y$ for one complete cycle of the point $x = 0$ on the string, we can analyze the equation given and then examine the images to match the characteristics.

The wave is described by the equation:

$y(x, t) = (1.2 \, \text{mm}) \sin((2.0 \pi \, \text{rad/s}) t - (0.50 \pi \, \text{rad/m}) x)$

Step-by-Step Analysis

1. Evaluate the Displacement at $x = 0$:

   • At $x = 0$, the equation simplifies to: $y(0, t) = (1.2 \, \text{mm}) \sin(2.0 \pi t)$
   • This is a simple sinusoidal function of $t$ with a frequency determined by the term $2.0 \pi$.

2. Determine the Frequency and Period:

   • The term $2.0 \pi$ represents the angular frequency $\omega$, so: $\omega = 2.0 \pi \, \text{rad/s}$
   • The period $T$ can be calculated using the formula $T = \frac{2 \pi}{\omega}$: $T = \frac{2 \pi}{2.0 \pi} = 1 \, \text{s}$
   • Therefore, the wave completes one cycle in 1 second.

3. Match the Graph with the Characteristics:

   • The displacement oscillates between $+1.2$ mm and $-1.2$ mm, with a period of 1 second for the point at $x = 0$.
   • Observing the graphs:
     • The first image shows a sinusoidal curve completing one cycle between $t = 0$ and $t = 2$ seconds.
     • The second image shows a sinusoidal curve completing one cycle between $t = 0$ and $t = 1$ second.

Since we expect one complete cycle over 1 second, the second graph is the correct one for the displacement $y$ at $x = 0$.

Would you like further details on wave properties or have any questions about this analysis?

Related Questions

1. How would the graph change if the amplitude were doubled?
2. What effect would changing the angular frequency have on the period?
3. How would the graph look if plotted for a different point $x \neq 0$?
4. Can you calculate the wavelength of the wave from the given information?
5. What would the graph look like if the phase constant was added?

Tip: For sinusoidal functions, the period is inversely proportional to the frequency—doubling the frequency halves the period.