HW 11 Simple Harmonic Motion and Pendulum | StudyX - Free Printable

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 Position:
- Velocity (\( v \)): Zero (the bob is momentarily at rest at the highest point).
- Kinetic Energy (\( KE \)): Zero (since \( KE = \frac{1}{2}mv^2 \) and \( v = 0 \)).
- Potential Energy (\( PE_g \)): Maximum (the bob is at the highest point).

##### Middle Position:
- Velocity (\( v \)): Maximum (the bob is moving fastest at the lowest point).
- Kinetic Energy (\( KE \)): Maximum (since \( KE = \frac{1}{2}mv^2 \) and \( v \) is maximum).
- Potential Energy (\( PE_g \)): Zero (the bob is at the lowest point, reference level).

##### Rightmost Position:
- Velocity (\( v \)): Zero (the bob is momentarily at rest at the highest point).
- Kinetic Energy (\( KE \)): Zero (since \( KE = \frac{1}{2}mv^2 \) and \( v = 0 \)).
- Potential Energy (\( PE_g \)): Maximum (the bob is at the highest point).

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

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

To determine the period experimentally:
1. Measure the length \( L \) of the pendulum (distance from the pivot to the center of mass of the bob).
2. Measure the time for multiple oscillations (e.g., 10 or 20 cycles) and divide by the number of cycles to get the period \( T \).

#### Table Setup:
The table requires recording the amplitude (in degrees) and the corresponding period \( T \) for different amplitudes. The mass and length of the pendulum should also be noted.

##### Example Data Collection:
Assume the following measurements:
- Mass of the bob: \( 0.1 \, \text{kg} \)
- Length of the pendulum: \( 1.0 \, \text{m} \)

| Amplitude (Degrees) | Period (s) |
|---------------------|------------|
| 5 | 2.0 |
| 10 | 2.0 |
| 15 | 2.0 |
| 20 | 2.0 |
| 25 | 2.0 |

#### Explanation:
- For small amplitudes, the period of a pendulum is independent of the amplitude (this is a characteristic of SHM).
- If the amplitude becomes very large, the period will slightly increase due to the nonlinearity of the restoring force.

Final Answer:

\[
\boxed{
\begin{array}{c|c|c}
\text{Position} & \text{Velocity (v)} & \text{Kinetic Energy (KE)} & \text{Potential Energy (PE_g)} \\
\hline
\text{Leftmost} & \text{Zero} & \text{Zero} & \text{Maximum} \\
\text{Middle} & \text{Maximum} & \text{Maximum} & \text{Zero} \\
\text{Rightmost} & \text{Zero} & \text{Zero} & \text{Maximum} \\
\end{array}
}
\]

For the period data, use the experimental values obtained from the PhET Pendulum Lab. If no specific values are provided, assume the period remains constant for small amplitudes.