3) The angle of pendulum from straight down will vary in a sinusoidal fashion as it swings. In Physics you will learn (if you haven't already) that for a frictionless pendulum, released k radians from feet vertical will be 0 (t) = k cos . g is gravitational acceleration, on Earth's surface ~32 s2 : •t For our problem the length is 20 inches, which will require conversion to be the Lin the equation. We will use k = 0.15 radians. a. Find the angular frequency (@) of the oscillation. L = 20 in. e = 0.15 rad b. Find the period in seconds. с. Find the time until (t) = 0. (ie when it is straight down)
3) The angle of pendulum from straight down will vary in a sinusoidal fashion as it swings. In Physics you will learn (if you haven't already) that for a frictionless pendulum, released k radians from feet vertical will be 0 (t) = k cos . g is gravitational acceleration, on Earth's surface ~32 s2 : •t For our problem the length is 20 inches, which will require conversion to be the Lin the equation. We will use k = 0.15 radians. a. Find the angular frequency (@) of the oscillation. L = 20 in. e = 0.15 rad b. Find the period in seconds. с. Find the time until (t) = 0. (ie when it is straight down)
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![**Pendulum Motion Analysis**
The angle of a pendulum from the vertical changes in a sinusoidal way as it swings. In physics, it is demonstrated that for a frictionless pendulum that is released with an angle \( k \) radians from the vertical, the angle \( \theta(t) \) is given by:
\[ \theta(t) = k \cos\left(\sqrt{\frac{g}{L}} \cdot t\right) \]
Where:
- \( g \) is the gravitational acceleration (approximately \( 32 \, \text{feet/s}^2 \) on Earth).
- \( L \) is the pendulum’s length.
For this specific analysis:
- The pendulum length \( L \) is 20 inches.
- \( k = 0.15 \) radians.
**Tasks:**
a. **Find the Angular Frequency (\( \omega \))**
Determine the expression for \( \omega \) as it relates to the gravitational acceleration and pendulum length.
b. **Find the Period in Seconds**
Calculate the period using the formula \( T = \frac{2\pi}{\omega} \).
c. **Time until \( \theta(t) = 0 \)**
Find when the pendulum is in the vertical position again, i.e., \( \theta = 0 \).
d. **Plot One Complete Cycle of the Waveform by Hand**
Illustrate the sine wave over one complete cycle, labeling the amplitude and period.
e. **Confirm Calculations Using Software**
Use a software tool to graph the function for one cycle. Compare this plot to your hand-drawn version for accuracy.
f. **Phase Offset for Sine Function**
Determine the phase offset needed if the equation used a sine function instead of cosine to observe the same pendulum behavior.
**Diagram Explanation:**
The image includes a diagram of a basic pendulum system, illustrating:
- The pendulum’s length (\( L = 20 \) in).
- The angle (\( \theta = 0.15 \) rad) from the vertical.
This educational content helps in understanding the dynamics of pendulum motion through mathematical representation and graphical visualization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88ce7fe7-6055-4d88-a2fe-fb962619b6d1%2F787bf2d6-e40a-438a-b53d-ac051abffdf5%2F2mrnyzp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Pendulum Motion Analysis**
The angle of a pendulum from the vertical changes in a sinusoidal way as it swings. In physics, it is demonstrated that for a frictionless pendulum that is released with an angle \( k \) radians from the vertical, the angle \( \theta(t) \) is given by:
\[ \theta(t) = k \cos\left(\sqrt{\frac{g}{L}} \cdot t\right) \]
Where:
- \( g \) is the gravitational acceleration (approximately \( 32 \, \text{feet/s}^2 \) on Earth).
- \( L \) is the pendulum’s length.
For this specific analysis:
- The pendulum length \( L \) is 20 inches.
- \( k = 0.15 \) radians.
**Tasks:**
a. **Find the Angular Frequency (\( \omega \))**
Determine the expression for \( \omega \) as it relates to the gravitational acceleration and pendulum length.
b. **Find the Period in Seconds**
Calculate the period using the formula \( T = \frac{2\pi}{\omega} \).
c. **Time until \( \theta(t) = 0 \)**
Find when the pendulum is in the vertical position again, i.e., \( \theta = 0 \).
d. **Plot One Complete Cycle of the Waveform by Hand**
Illustrate the sine wave over one complete cycle, labeling the amplitude and period.
e. **Confirm Calculations Using Software**
Use a software tool to graph the function for one cycle. Compare this plot to your hand-drawn version for accuracy.
f. **Phase Offset for Sine Function**
Determine the phase offset needed if the equation used a sine function instead of cosine to observe the same pendulum behavior.
**Diagram Explanation:**
The image includes a diagram of a basic pendulum system, illustrating:
- The pendulum’s length (\( L = 20 \) in).
- The angle (\( \theta = 0.15 \) rad) from the vertical.
This educational content helps in understanding the dynamics of pendulum motion through mathematical representation and graphical visualization.
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