A mass attached to the end of a spring is oscillating with a period of 2.25 s on a horizontal frictionless surface. The mass was released from rest at t = 0 from the position x = 0.0580 m.
Determine the location of the mass at t = 5.47 s?
How do I solve this? Do not give me the answer.

To solve this, we need to consider that a mass on a spring in simple harmonic motion (SHM) follows a specific form of the oscillatory equation. Since the mass is released from rest, we can use the position function for SHM:

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

where:
- \( x(t) \) is the position at time \( t \),
- \( A \) is the amplitude (maximum displacement from the equilibrium position),
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant, which we determine based on initial conditions.

### Step 1: Analyze the Initial Conditions
At \( t = 0 \), the mass is at \( x = 0.0580 \, \text{m} \) and released from rest, meaning it starts at maximum displacement with no initial velocity.

From this:
- The amplitude \( A \) is 0.0580 m.
- Since it's at maximum displacement at \( t = 0 \), the cosine function starts at its peak, meaning \( \phi = 0 \).

### Step 2: Calculate Angular Frequency \( \omega \)
The period \( T \) is given as 2.25 s. Angular frequency \( \omega \) is related to the period by:

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

### Step 3: Write the Position Equation
Substitute \( A \), \( \omega \), and \( \phi \) into the position function:

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

### Step 4: Substitute \( t = 5.47 \, \text{s} \) to Find the Position
Now, plug \( t = 5.47 \) s into the equation to determine \( x(5.47) \).

Would you like guidance on calculating each of these steps, or do you have any questions? 

---

1. How does changing the initial conditions affect the position function?
2. What would be different if the mass had an initial velocity?
3. How would you find the velocity of the mass at \( t = 5.47 \) s?
4. How can you verify that this solution satisfies the equations of motion?
5. What effect would adding friction have on the oscillation?

**Tip:** For oscillatory motion, always double-check the phase constant, as it depends on initial conditions.

A mass attached to the end of a spring is oscillating with a period of 2.25 s on a horizontal frictionless surface. The mass was released from rest at t = 0 from the position x = 0.0580 m. Determine the location of the mass at t = 5.47 s?

Learn how to determine the position of a mass oscillating on a spring with a period of 2.25 seconds. This detailed solution explains how to apply the simple harmonic motion equation to find the location of the mass at t = 5.47 s. Suitable for high school students studying physics and trigonometry.

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