College Physics
11th Edition
ISBN: 9781305952300
Author: Raymond A. Serway, Chris Vuille
Publisher: Cengage Learning
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![### Problem Statement
In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression:
\[ x = 7.00 \cos \left( 3t + \frac{\pi}{4} \right) \]
where \( x \) is in centimeters and \( t \) is in seconds.
#### Tasks
(a) At \( t = 0 \), find the position of the piston.
\[ \text{Position:} \quad \underline{\qquad\qquad} \, \text{cm} \]
(b) At \( t = 0 \), find the velocity of the piston.
\[ \text{Velocity:} \quad \underline{\qquad\qquad} \, \text{cm/s} \]
(c) At \( t = 0 \), find the acceleration of the piston.
\[ \text{Acceleration:} \quad \underline{\qquad\qquad} \, \text{cm/s}^2 \]
(d) Find the period and amplitude of the motion.
\[ \text{Period:} \quad \underline{\qquad\qquad} \, \text{s} \]
\[ \text{Amplitude:} \quad \underline{\qquad\qquad} \, \text{cm} \]
### Explanation of Concepts
- **Position**: The location of the piston at a specific time.
- **Velocity**: The rate of change of position of the piston with respect to time.
- **Acceleration**: The rate of change of velocity with respect to time.
- **Amplitude**: The maximum extent of the oscillation, which is the coefficient of the cosine function.
- **Period**: The time taken for one complete cycle of the motion, calculated as \( \frac{2\pi}{\text{angular frequency}} \).
By solving these equations, students can understand how harmonic motion describes the movement of oscillating systems like a piston.](https://content.bartleby.com/qna-images/question/56b9ddaf-a5b9-4606-96af-f9c7010351c4/878afc24-4f59-489f-b4ae-b5fb8a9f3645/4bmiet_processed.png)
Transcribed Image Text:### Problem Statement
In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression:
\[ x = 7.00 \cos \left( 3t + \frac{\pi}{4} \right) \]
where \( x \) is in centimeters and \( t \) is in seconds.
#### Tasks
(a) At \( t = 0 \), find the position of the piston.
\[ \text{Position:} \quad \underline{\qquad\qquad} \, \text{cm} \]
(b) At \( t = 0 \), find the velocity of the piston.
\[ \text{Velocity:} \quad \underline{\qquad\qquad} \, \text{cm/s} \]
(c) At \( t = 0 \), find the acceleration of the piston.
\[ \text{Acceleration:} \quad \underline{\qquad\qquad} \, \text{cm/s}^2 \]
(d) Find the period and amplitude of the motion.
\[ \text{Period:} \quad \underline{\qquad\qquad} \, \text{s} \]
\[ \text{Amplitude:} \quad \underline{\qquad\qquad} \, \text{cm} \]
### Explanation of Concepts
- **Position**: The location of the piston at a specific time.
- **Velocity**: The rate of change of position of the piston with respect to time.
- **Acceleration**: The rate of change of velocity with respect to time.
- **Amplitude**: The maximum extent of the oscillation, which is the coefficient of the cosine function.
- **Period**: The time taken for one complete cycle of the motion, calculated as \( \frac{2\pi}{\text{angular frequency}} \).
By solving these equations, students can understand how harmonic motion describes the movement of oscillating systems like a piston.
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