An object is attached to a coiled spring. It is pulled down a distance of 3 inches from its equilibrium position, then released.

If the time for one complete oscillation is 7 seconds, find the equation of the simple harmonic motion.

To find the equation of the simple harmonic motion, we need to determine the form of the motion and identify the parameters involved.

For simple harmonic motion, the displacement \( x(t) \) from the equilibrium position can be modeled as:

\[ x(t) = A \cos(\omega t + \phi) \]

where:
- \( A \) is the amplitude (the maximum displacement from the equilibrium position).
- \( \omega \) is the angular frequency.
- \( t \) is the time.
- \( \phi \) is the phase constant.

Given dataTo find the equation of the simple harmonic motion, we need to determine the form of the motion and identify the parameters involved.

For simple harmonic motion, the displacement \( x(t) \) from the equilibrium position can be modeled as:

\[ x(t) = A \cos(\omega t + \phi) \]

where:
- \( A \) is the amplitude (the maximum displacement from the equilibrium position).
- \( \omega \) is the angular frequency.
- \( t \) is the time.
- \( \phi \) is the phase constant.

Given 
1. The amplitude \( A \) is 3 inches (since it is pulled down 3 inches from the equilibrium).
2. The period \( T \) is 7 seconds (time for one complete oscillation).

We start by calculating the angular frequency \( \omega \). The relationship between the period \( T \) and the angular frequency \( \omega \) is given by:

\[ \omega = \frac{2\pi}{T} \]

Substituting the given period \( T = 7 \) seconds:

\[ \omega = \frac{2\pi}{7} \]

Next, we need to determine the phase constant \( \phi \). Since the object is pulled down and then released, it starts at maximum displacement. This corresponds to a cosine function with no phase shift if we start measuring time from the moment of release. Hence, \( \phi = 0 \).

Thus, the equation of motion becomes:

\[ x(t) = 3 \cos\left(\frac{2\pi}{7} t\right) \]

In summary, the equation of the simple harmonic motion is:

\[ x(t) = 3 \cos\left(\frac{2\pi}{7} t\right) \]

Would you like any further details or have any questions?

Here are some questions you might ask next:
1. How do we determine the velocity and acceleration of the object in simple harmonic motion?
2. Can you explain the energy changes in simple harmonic motion?
3. What would happen if the period of oscillation were to change?
4. How does damping affect the equation of simple harmonic motion?
5. Can we derive the equation of motion using differential equations?

**Tip:** Remember that in simple harmonic motion, the restoring force is always proportional to the displacement and acts in the opposite direction.

Discover how to find the equation of simple harmonic motion when an object with an amplitude of 3 inches oscillates with a period of 7 seconds. Learn about the concepts of amplitude, angular frequency, and phase constant in this high school-level physics problem.

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