The graph shows the position $x(t)$ of a mass undergoing simple harmonic motion (SHM) as a function of time. We are asked to determine the initial phase constant $\phi_0$, with the answer to be expressed in radians between $-1.57$ and $1.57$ (i.e., between $-\pi/2$ and $\pi/2$).

Steps to Determine the Phase Constant:

The general equation for SHM is given by: $x(t) = A \cos(\omega t + \phi_0)$ where:

• $A$ is the amplitude of the motion,
• $\omega$ is the angular frequency,
• $\phi_0$ is the phase constant.

1. Amplitude: From the graph, the maximum displacement (amplitude) appears to be around $A = 6 \, \text{cm}$.

2. Angular Frequency $\omega$: The period $T$ of the oscillation is the time it takes for one full cycle. Based on the graph:

   • The time for one cycle is approximately $T = 0.8 \, \text{s}$.
   • The angular frequency $\omega$ is given by: $\omega = \frac{2\pi}{T} = \frac{2\pi}{0.8} \approx 7.85 \, \text{rad/s}$

3. Initial Phase $\phi_0$:

   • At $t = 0$, the position $x(0) \approx 6 \, \text{cm}$ (near the maximum displacement).
   • Since the object starts near the maximum displacement and not at equilibrium, we can infer that the motion corresponds to a cosine function. When $t = 0$, $x(0) \approx A$, which suggests $\cos(\phi_0) = 1$, meaning $\phi_0 = 0$.

Therefore, the initial phase constant $\phi_0$ is approximately: $\boxed{0 \, \text{rad}}$

Would you like further clarification or more details on any step?

Related Questions:

1. How does changing the amplitude affect the initial phase constant?
2. What is the relationship between the angular frequency and the period of oscillation?
3. If the object started at the equilibrium point, what would the phase constant be?
4. How would you determine the phase constant if the graph started at a minimum position?
5. How does damping affect the equation for simple harmonic motion?

Tip: In simple harmonic motion, the initial phase constant helps determine where the object is at $t = 0$.