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Here’s a breakdown of the problem:

The displacement of a spring-mass system is given by \( x = A \cos(\omega t + \phi) \), where:

- \( A \) is the amplitude.
- \( \omega \) is the angular frequency.
- \( \phi \) is the phase angle.
- \( t \) is time.

### Explanation:
1. The phase angle \( \phi \) determines the initial state of the system at \( t = 0 \).
2. At \( t = 0 \):
   \[
   x = A \cos(\phi)
   \]
   This equation tells us the initial displacement.

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### Answers:
#### (a) What angle(s) of \( \phi \) (in radians) describe a spring initially fully compressed?
- The spring is fully compressed when \( x = -A \), meaning:
  \[
  A \cos(\phi) = -A \implies \cos(\phi) = -1
  \]
  - The angle \( \phi = \pi \) radians (or any odd multiple of \( \pi \), e.g., \( 3\pi, -\pi \)) describes this state.

#### (b) What angle(s) of \( \phi \) (in radians) describe a mass initially at its maximum velocity?
- The mass is at maximum velocity when the displacement is zero (\( x = 0 \)) because velocity \( v = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi) \) is maximized at equilibrium position.
  \[
  A \cos(\phi) = 0 \implies \cos(\phi) = 0
  \]
  - The angle \( \phi = \frac{\pi}{2} \) or \( \phi = \frac{3\pi}{2} \) radians describes this state.

#### (c) How to differentiate them physically?
- When \( \phi = \pi \) (fully compressed), the spring is at its maximum displacement in the negative direction, and the mass is momentarily stationary (\( v = 0 \)).
- When \( \phi = \frac{\pi}{2} \) or \( \phi = \frac{3\pi}{2} \), the spring passes through equilibrium, and the mass is moving at its fastest speed in either direction (\( +v_{\text{max}} \) or \(-v_{\text{max}} \)).

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### Additional Questions:
1. What happens when \( \phi = 0 \) and \( t = 0 \)? What does it describe physically?
2. Why is the velocity maximized when displacement is zero in harmonic motion?
3. How does changing \( \phi \) affect the phase shift in the cosine graph?
4. What is the significance of angular frequency \( \omega \) in this equation?
5. How would the answers change if the equation were sine-based, \( x = A \sin(\omega t + \phi) \)?

### Tip:
Visualizing this system on a graph (like Desmos) can help you see how \( \phi \) shifts the motion along

Suppose x = A cos(ωt + ϕ). When ϕ = 0, we mean to describe a system that initially has a fully stretched spring (and mass not moving). (a) What angle/s of ϕ (in radians) describe a spring initially fully compressed? (b) What angle/s of ϕ (in radians) describe a mass initially at its maximum velocity? (c) In your answer in letter b), how would you differentiate them physically?

Explore how to calculate phase angles for a spring-mass system using the formula x = A cos(ωt + ϕ). Learn the conditions for fully compressed springs and maximum velocity states, along with their physical interpretations.

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