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This is from a physics textbook. Solve it.

To solve this problem, we need to find the acceleration of a point on the blue curve at \( t = 1 \) second.

### Steps to Solve:
1. **Understand the Graph**:
   - The blue curve represents a sinusoidal function, and the \( x \)-axis represents time \( t \) in seconds.
   - The vertical axis (displacement in cm) shows how the position \( x(t) \) changes over time.

2. **Determine the Function**:
   - The blue curve appears to be a sinusoidal wave. A general form for such a wave is:
     \[
     x(t) = A \cos(\omega t + \phi)
     \]
     where:
     - \( A \) is the amplitude,
     - \( \omega \) is the angular frequency,
     - \( \phi \) is the phase shift.
   - From the graph:
     - The amplitude \( A \) is \( 4 \) cm (maximum displacement from equilibrium).
     - The period \( T \) of the wave is \( 2 \) seconds (time between successive peaks or troughs).
     - \( \omega = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi \) rad/s.

3. **Find the Velocity and Acceleration**:
   - The velocity \( v(t) \) is the first derivative of position \( x(t) \):
     \[
     v(t) = \frac{dx(t)}{dt} = -A\omega \sin(\omega t + \phi)
     \]
   - The acceleration \( a(t) \) is the derivative of velocity, or the second derivative of position:
     \[
     a(t) = \frac{dv(t)}{dt} = -A\omega^2 \cos(\omega t + \phi)
     \]
   - Substituting the values \( A = 4 \) cm and \( \omega = \pi \) rad/s into the acceleration formula:
     \[
     a(t) = -4\pi^2 \cos(\pi t + \phi)
     \]

4. **Evaluate at \( t = 1 \) second**:
   - At \( t = 1 \) s, examine the cosine term in the acceleration formula:
     \[
     a(1) = -4\pi^2 \cos(\pi \times 1 + \phi)
     \]
     \[
     a(1) = -4\pi^2 \cos(\pi + \phi)
     \]
   - From the graph at \( t = 0 \), the blue curve starts at a positive peak, suggesting that \( \phi = 0 \) or \( \phi = 2\pi \).
     \[
     a(1) = -4\pi^2 \cos(\pi) = -4\pi^2 \times (-1) = 4\pi^2 \text{ cm/s}^2
     \]

### Final Answer:
The acceleration of a point on the blue curve at \( t = 1 \) second is \( 4\pi^2 \) cm/s².

Would you like more details on any step, or do you have any questions?

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Here are 5 related questions to further your understanding:
1. What is the relationship between displacement, velocity, and acceleration in sinusoidal motion?
2. How does the phase shift \( \phi \) affect the shape of the sinusoidal wave?
3. What is the significance of angular frequency \( \omega \) in periodic motion?
4. How would the acceleration change if the amplitude \( A \) were doubled?
5. What would the graph of the acceleration as a function of time look like for the blue curve?

**Tip:** Remember that the acceleration in simple harmonic motion is always directed towards the equilibrium position and is proportional to the displacement.

Learn how to solve a sinusoidal motion problem involving finding acceleration at t = 1 second. Detailed step-by-step solution using sinusoidal motion formulas. Suitable for advanced high school and college-level students.

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