To find the maximum acceleration in the simple harmonic motion described, we follow these steps:

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1. Expression for acceleration in SHM

The general expression for acceleration $a(t)$ in SHM is:
$a(t) = -\omega^2 x(t)$
where $\omega$ is the angular frequency and $x(t)$ is the displacement.

The velocity function $v(t)$ is given as:
$v(t) = 8.00 \, \text{cm/s} \, \sin(\omega t) + 10.0 \, \text{cm/s} \, \cos(\omega t)$

The acceleration is the time derivative of $v(t)$:
$a(t) = \frac{d}{dt} \left[ v(t) \right]$

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2. Calculate $a(t)$ using the derivative

Differentiating $v(t)$:
$a(t) = \frac{d}{dt} \left[ 8.00 \sin(\omega t) + 10.0 \cos(\omega t) \right]$
Using $\frac{d}{dt}[\sin(\omega t)] = \omega \cos(\omega t)$ and $\frac{d}{dt}[\cos(\omega t)] = -\omega \sin(\omega t)$:
$a(t) = 8.00 \omega \cos(\omega t) - 10.0 \omega \sin(\omega t)$
$a(t) = \omega \left[ 8.00 \cos(\omega t) - 10.0 \sin(\omega t) \right]$

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3. Maximum acceleration $|a_{\text{max}}|$

The maximum acceleration occurs when the expression inside the brackets reaches its maximum amplitude. This is the amplitude of the function:
$8.00 \cos(\omega t) - 10.0 \sin(\omega t)$
The amplitude is found as:
$A = \sqrt{(8.00)^2 + (-10.0)^2} = \sqrt{64 + 100} = \sqrt{164} = 12.81 \, \text{cm/s}$

Thus, the maximum acceleration is:
$|a_{\text{max}}| = \omega \cdot A$

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4. Find angular frequency $\omega$

The angular frequency $\omega$ is related to the spring constant $k$ and the mass $m$:
$\omega = \sqrt{\frac{k}{m}}$
Given $k = 9.00 \, \text{N/m}$ and $m = 4.00 \, \text{kg}$:
$\omega = \sqrt{\frac{9.00}{4.00}} = \sqrt{2.25} = 1.50 \, \text{rad/s}$

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5. Calculate $|a_{\text{max}}|$:

$|a_{\text{max}}| = \omega \cdot A = (1.50) \cdot (12.81) = 19.22 \, \text{cm/s}^2$

Convert to SI units (m/s²):
$|a_{\text{max}}| = 0.1922 \, \text{m/s}^2$

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Final Answer:

The maximum acceleration is:
$\boxed{0.192 \, \text{m/s}^2}$

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Do you need further clarification or details?

Relative Questions:

1. How is the angular frequency related to the physical properties of the spring system?
2. What would change if the spring constant were doubled?
3. How does the phase difference affect the velocity and acceleration functions in SHM?
4. What is the relationship between displacement, velocity, and acceleration in SHM?
5. How does the mass of the object affect the frequency of the motion?

Tip:

Always analyze the amplitude and angular frequency separately when determining the extremes of SHM quantities.