The problem provides a pendulum with its angular displacement given by the equation:

$\theta(t) = 0.18 \cos(9t)$

where:

• $\theta(t)$ is the angular displacement in radians,
• $t$ is time in seconds.

We need to determine:

1. The maximum angular displacement ($\theta_{\text{max}}$),
2. The rate of change of $\theta$ ($\theta'(t)$) at $t = 5$ seconds.

1. Maximum Angular Displacement ($\theta_{\text{max}}$)

The maximum angular displacement is simply the amplitude of the cosine function. Since the amplitude of $\cos(9t)$ is $1$, multiplying it by $0.18$ gives:

$\theta_{\text{max}} = 0.18 \, \text{radians}.$

2. Rate of Change of $\theta$ at $t = 5$

To find $\theta'(t)$, we differentiate $\theta(t) = 0.18 \cos(9t)$ with respect to $t$:

$\theta'(t) = -0.18 \cdot 9 \sin(9t).$

$\theta'(t) = -1.62 \sin(9t).$

At $t = 5$:

$\theta'(5) = -1.62 \sin(9 \cdot 5).$

Simplify $9 \cdot 5 = 45$, and find $\sin(45)$:

$\sin(45) \approx 0.8509.$

So:

$\theta'(5) = -1.62 \cdot 0.8509 \approx -1.378 \, \text{rad/sec}.$

Final Answers:

1. $\theta_{\text{max}} = 0.180 \, \text{radians}$,
2. $\theta'(5) = -1.378 \, \text{rad/sec}$.

Let me know if you'd like further clarifications or a step-by-step derivation.

Here are 5 related questions for practice:

1. What is the period of the pendulum's oscillation based on the given equation?
2. How would $\theta(t)$ change if the frequency were doubled?
3. What is the velocity of the pendulum when $\theta(t)$ reaches its maximum?
4. How can you determine the maximum angular velocity of the pendulum?
5. How does the length of the pendulum (27 cm) affect its frequency?

Tip: Understanding the relationship between angular displacement, velocity, and acceleration in harmonic motion is key to solving pendulum problems!