To calculate the acceleration and G-force of the given trigonometric function $y = a \cos \left( bx + \left( 30 - \frac{\pi}{2} \right) \right) + 20$, we’ll need to delve into its second derivative with respect to $x$. Acceleration in this context is often the second derivative $y''$, and G-force is typically the acceleration normalized by the gravitational constant $g \approx 9.81 \, \text{m/s}^2$.

Step-by-Step Solution:

1. Determine the First Derivative (Velocity): $y' = -a b \sin \left( bx + \left( 30 - \frac{\pi}{2} \right) \right)$

2. Calculate the Second Derivative (Acceleration): $y'' = -a b^2 \cos \left( bx + \left( 30 - \frac{\pi}{2} \right) \right)$ Here, $y''$ gives the acceleration.

3. Substitute $a$ and $b$ Values: Using the provided values $a = \frac{30}{\pi + 15}$ and $b = \frac{\pi}{15} + 1$, we can substitute these into $y''$ to get a specific acceleration expression.

4. Calculate G-force: $\text{G-force} = \frac{|y''|}{g}$

Would you like a detailed calculation of $y''$ with specific values, or do you have another question?

Related Questions:

1. How does changing $a$ affect the acceleration amplitude?
2. What impact does varying $b$ have on the frequency of oscillations?
3. How can the turning point be used to verify $a$ and $b$?
4. What is the significance of the phase shift $\left( 30 - \frac{\pi}{2} \right)$?
5. How would this function behave if plotted over a large interval?

Tip: The G-force is usually highest at the points of maximum acceleration in a sinusoidal function.