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The problem describes a pendulum that completes two cycles every 1.3 seconds. It starts from rest at an angle of \( 12^\circ \) to the vertical, and we are tasked with determining the position (angle in radians) of the pendulum at \( t = 0.1 \, \text{s} \) and \( t = 2.15 \, \text{s} \), ignoring friction.

Let's break it down:

### Step 1: Angular Frequency and Period
Since the pendulum completes **two cycles** every **1.3 seconds**, the frequency is:
\[
f = \frac{2}{1.3} \, \text{Hz}
\]
The angular frequency \( \omega \) is related to the frequency by:
\[
\omega = 2 \pi f
\]

### Step 2: Pendulum Equation
The position of the pendulum as a function of time, assuming simple harmonic motion, is given by:
\[
\theta(t) = \theta_0 \cos(\omega t)
\]
where:
- \( \theta_0 = 12^\circ \) is the maximum angular displacement (initial angle),
- \( \omega \) is the angular frequency, and
- \( t \) is the time.

We will convert \( \theta_0 \) from degrees to radians and substitute the values to compute the pendulum's position at the specified times.

Let me calculate these steps.The positions (angles in radians) of the pendulum are:

- At \( t = 0.1 \, \text{s} \): \( 0.119 \, \text{rad} \)
- At \( t = 2.15 \, \text{s} \): \( -0.074 \, \text{rad} \)

These results describe the pendulum's displacement from its vertical equilibrium position at the specified times.

Would you like further explanation on any part of the process? Here are five related questions for deeper exploration:

1. How is the period of a pendulum related to its frequency?
2. What is the significance of angular frequency in simple harmonic motion?
3. Why do we use radians instead of degrees in pendulum calculations?
4. How would the presence of friction affect the pendulum's motion over time?
5. What would the graph of \( \theta(t) \) look like over a complete cycle?

**Tip:** Always ensure that angles are converted to radians when using trigonometric functions in physical equations.

A clock pendulum completes 2 cycles every 1.3 seconds. It is released from rest starting at an angle of 12° to the vertical. Ignoring friction, determine the position (angle in radians) of the pendulum at t = 0.1 s and t = 2.15 s.

This physics problem calculates the angular position of a pendulum after it completes two cycles every 1.3 seconds. Learn how to find the position at two different times using angular frequency and trigonometry.

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