Pendulum oscillation diagram illustrating energy and velocity changes at different points in the swing.

Problem Analysis:

The homework assignment focuses on understanding the concepts of simple harmonic motion (SHM) and the behavior of a pendulum. The task involves analyzing the energy and velocity of a pendulum at different points in its oscillation and determining the period of the pendulum under varying conditions.

#### Key Concepts:
1. Simple Harmonic Motion (SHM):
- SHM is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction of the displacement.
- For a pendulum, the motion is approximately SHM for small angles.

2. Energy in a Pendulum:
- Kinetic Energy (KE): Maximum when the pendulum bob is at the lowest point (velocity is maximum).
- Potential Energy (PE): Maximum when the pendulum bob is at the highest point (velocity is zero).
- Total mechanical energy is conserved if friction is negligible.

3. Period of a Pendulum:
- The period \( T \) of a simple pendulum is given by:
\[
T = 2\pi \sqrt{\frac{L}{g}}
\]
where:
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

Solution:

#### Part 1: Labeling the Diagram
The diagram shows three positions of the pendulum bob during its oscillation:
1. Leftmost Position: Highest point on the left.
2. Middle Position: Lowest point (equilibrium position).
3. Rightmost Position: Highest point on the right.

We need to label the velocity (\( v \)), kinetic energy (\( KE \)), and gravitational potential energy (\( PE_g \)) at each position.

##### Leftmost and Rightmost Positions:
- Velocity (\( v \)): Zero because the pendulum bob momentarily stops at the highest points.
- Kinetic Energy (\( KE \)): Zero because \( KE = \frac{1}{2}mv^2 \), and \( v = 0 \).
- Potential Energy (\( PE_g \)): Maximum because the bob is at the highest point, and \( PE_g = mgh \), where \( h \) is the height above the equilibrium position.

##### Middle Position:
- Velocity (\( v \)): Maximum because the bob is moving fastest at the lowest point.
- Kinetic Energy (\( KE \)): Maximum because \( KE = \frac{1}{2}mv^2 \), and \( v \) is maximum.
- Potential Energy (\( PE_g \)): Zero (or minimum) because the bob is at the equilibrium position, and \( h = 0 \).

#### Filled Diagram:
```
v = zero v = zero
KE = zero KE = zero
PE_g = maximum PE_g = maximum

v = maximum
KE = maximum
PE_g = zero
```

#### Part 2: Determining the Period of the Pendulum
The period \( T \) of a pendulum depends on its length \( L \) and the acceleration due to gravity \( g \). The formula is:
\[
T = 2\pi \sqrt{\frac{L}{g}}
\]

To determine the period under varying conditions, you need to measure the time it takes for the pendulum to complete one full oscillation (from one extreme to the other and back) for different amplitudes. The amplitude should be small (less than about 15 degrees) for the motion to approximate SHM.

##### Steps:
1. Measure the length \( L \) of the pendulum (distance from the pivot to the center of mass of the bob).
2. Set the amplitude (angle from the vertical) and release the pendulum.
3. Use a stopwatch to measure the time for 10 or more oscillations to improve accuracy, then divide by the number of oscillations to get the period \( T \).
4. Repeat for different amplitudes and record the data.

##### Example Data Table:
| Amplitude (Degrees) | Period (s) |
|---------------------|------------|
| 5 | |
| 10 | |
| 15 | |
| 20 | |
| 25 | |

#### Explanation:
- For small amplitudes (e.g., 5° to 15°), the period \( T \) should remain nearly constant because the motion is approximately SHM.
- For larger amplitudes (e.g., 20° to 25°), the period may increase slightly because the approximation of SHM breaks down.

Final Answer:

\[
\boxed{
\begin{array}{c|c|c}
\text{Position} & \text{Velocity (v)} & \text{Kinetic Energy (KE)} & \text{Potential Energy (PE}_g\text{)} \\
\hline
\text{Leftmost} & \text{zero} & \text{zero} & \text{maximum} \\
\text{Middle} & \text{maximum} & \text{maximum} & \text{zero} \\
\text{Rightmost} & \text{zero} & \text{zero} & \text{maximum} \\
\end{array}
}
\]

For the period determination, follow the steps outlined above and fill in the table with your measured values.