In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression, (3t+1) x = 7.00 cos (3t+ where x is in centimeters and t is in seconds. (a) At t = 0, find the position of the piston. cm (b) At t = 0, find velocity of the piston. cm/s (c) At t = 0, find acceleration of the piston. cm/s²
Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
![### 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56b9ddaf-a5b9-4606-96af-f9c7010351c4%2F878afc24-4f59-489f-b4ae-b5fb8a9f3645%2F4bmiet_processed.png&w=3840&q=75)
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