I see the worksheet you've provided involves problems requiring mathematical modeling and analysis based on physics equations, particularly related to motion and forces. Here's an explanation and solution approach for each question visible:

Question 5:

Task: Determine expressions for $y_{\text{max}}, v_{\text{max}},$ and $a_{\text{max}}$. Analyze where these values occur in the cycle.

• Given: Equation $y = 0.003479 \cos(8.143t + 5.667) + 0.4594$.

Steps:

1. $y_{\text{max}}$ occurs when $\cos(8.143t + 5.667) = 1$.

   • Maximum value: $y_{\text{max}} = 0.003479 \cdot 1 + 0.4594 = 0.462879$.

2. $v_{\text{max}}$:

   • Velocity $v(t) = \frac{dy}{dt}$.
   • From the equation $v(t) = -0.028329 \sin(8.143t + 5.667)$, the maximum value occurs when $\sin(8.143t + 5.667) = \pm 1$:
   • $v_{\text{max}} = 0.028329$.

3. $a_{\text{max}}$:

   • Acceleration $a(t) = \frac{d^2y}{dt^2}$.
   • From the equation $a(t) = -0.230697 \cos(8.143t + 5.667)$, the maximum value occurs when $\cos(8.143t + 5.667) = \pm 1$:
   • $a_{\text{max}} = 0.230697$.

Question 6:

Task: Write an equation for maximum acceleration and calculate angular frequency $\omega$ using Newton's Second Law.

• Newton's Second Law: $F = ma$, where $a = \omega^2 A$.
• Maximum acceleration: $a_{\text{max}} = \omega^2 A$.
  • Using $A = 0.003479$ and $a_{\text{max}} = 0.230697$:
    • $\omega^2 = \frac{a_{\text{max}}}{A} = \frac{0.230697}{0.003479} \approx 66.3$.
    • $\omega = \sqrt{66.3} \approx 8.143 \ \text{rad/s}$ (consistent with given data).

Question 7:

Task: Calculate the period $T$ using $T = \frac{2\pi}{\omega}$.

• Given: $\omega = 8.143 \ \text{rad/s}$.
• Period: $T = \frac{2\pi}{8.143} \approx 0.771 \ \text{s}$.

Comparison with Graphs:

• Check if the period matches the graph's periodicity. If consistent, this validates the model.

Would you like detailed derivations, or do you have further questions? Here are some relative queries:

1. How is the period $T$ connected to angular frequency $\omega$?
2. What is the significance of finding $y_{\text{max}}, v_{\text{max}}, a_{\text{max}}$?
3. How does Newton's Second Law apply to harmonic motion?
4. What would happen to the period if $\omega$ were doubled?
5. How do phase shifts (e.g., $+5.667$) affect the motion's maximum values?

Tip: Always relate angular frequency $\omega$ to measurable parameters like period $T$ and maximum velocity/acceleration for deeper insights.