To model the position of the object attached to the coiled spring, we can use the equation for simple harmonic motion (SHM). The general form for the position of an object undergoing SHM is:

$s(t) = A \cos(\omega t + \phi)$

Where:

• $A$ is the amplitude (the maximum displacement from the equilibrium position),
• $\omega$ is the angular frequency (which relates to the period of the motion),
• $t$ is the time,
• $\phi$ is the phase constant, which determines the initial condition of the motion.

Step 1: Identify the given information

• Amplitude $A = 6$ (since the object is pulled down 6 units from equilibrium),
• Period $T = 5$ seconds (time for one complete oscillation).

Step 2: Calculate the angular frequency ($\omega$)

The angular frequency $\omega$ is related to the period $T$ by the formula:

$\omega = \frac{2\pi}{T}$

Substitute the value of $T$:

$\omega = \frac{2\pi}{5}$

Step 3: Determine the phase constant ($\phi$)

Since the object is released from the maximum displacement (6 units down), the initial position at $t = 0$ is $-6$ (assuming downward is negative). This suggests that the cosine function starts at its maximum displacement (negative). Hence, the phase constant is $\phi = \pi$, which reflects this downward displacement.

Step 4: Write the equation

Now that we have all the components, the equation for the position of the object at time $t$ is:

$s(t) = 6 \cos\left(\frac{2\pi}{5} t + \pi\right)$

Final Answer:

$s(t) = 6 \cos\left(\frac{2\pi}{5} t + \pi\right)$

Would you like further details or explanations on any part of this? Here are some related questions to explore:

1. How do the concepts of amplitude and angular frequency relate to the motion of a spring?
2. How would the equation change if the object were released from a different position?
3. What is the difference between the period and frequency of oscillation?
4. How can you calculate the velocity or acceleration of the object at a given time?
5. What factors affect the period of a spring's oscillation?

Tip: Remember that the phase constant $\phi$ depends on the initial displacement and velocity. If the object were initially at rest, this could affect the form of the equation.