The given function models the displacement of an object moving in simple harmonic motion. y = 6 sin(2t) (a) Find the amplitude, period, and frequency of the motion. amplitude period frequency (b) Sketch a graph of the displacement of the object over one complete period.
The given function models the displacement of an object moving in simple harmonic motion. y = 6 sin(2t) (a) Find the amplitude, period, and frequency of the motion. amplitude period frequency (b) Sketch a graph of the displacement of the object over one complete period.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The given function models the displacement of an object moving in simple harmonic motion.
y = 6 sin(2t)
(a) Find the amplitude, period, and frequency of the motion.
(b) Sketch a graph of the displacement of the object over one complete period.
| amplitude |
|
| period |
|
| frequency |
|
(b) Sketch a graph of the displacement of the object over one complete period.
![The given function models the displacement of an object moving in simple harmonic motion.
\[ y = 6 \sin(2t) \]
(a) Find the amplitude, period, and frequency of the motion.
- Amplitude: \_\_\_\_
- Period: \_\_\_\_
- Frequency: \_\_\_\_
(b) Sketch a graph of the displacement of the object over one complete period.
**Graphs Explanation:**
Four versions of the sinusoidal graph are displayed, each representing the function \( y = 6 \sin(2t) \), and two for the normalized sine function \(\sin(2t)\):
1. **First and Fourth Graphs:**
- Axis: \(y\) ranges from \(-6\) to \(6\) and \(t\) extends from \(0\) to \(\pi\).
- The sine wave starts at the origin, peaks at \(y = 6\) at \(t = \frac{\pi}{4}\), crosses back through zero at \(t = \frac{\pi}{2}\), reaches a minimum of \(y = -6\) at \(t = \frac{3\pi}{4}\), and returns to zero at \(t = \pi\).
2. **Second and Third Graphs:**
- Axis: \(y\) ranges from \(-1\) to \(1\) and \(t\) extends from \(0\) to \(\pi\).
- Similar to the scaled version, but reflects a standard sine wave for \(\sin(2t)\) without amplitude scaling. Peaks and troughs are at \(1\) and \(-1\) respectively.
Each sine graph follows the typical pattern of oscillation, illustrating one complete cycle over the interval from \(0\) to \(\pi\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50b7307d-f147-4c39-812e-01e48cbfa721%2Fd48170a0-2838-4dd2-af44-c0c3855f7e84%2Faj74g1g_processed.png&w=3840&q=75)
Transcribed Image Text:The given function models the displacement of an object moving in simple harmonic motion.
\[ y = 6 \sin(2t) \]
(a) Find the amplitude, period, and frequency of the motion.
- Amplitude: \_\_\_\_
- Period: \_\_\_\_
- Frequency: \_\_\_\_
(b) Sketch a graph of the displacement of the object over one complete period.
**Graphs Explanation:**
Four versions of the sinusoidal graph are displayed, each representing the function \( y = 6 \sin(2t) \), and two for the normalized sine function \(\sin(2t)\):
1. **First and Fourth Graphs:**
- Axis: \(y\) ranges from \(-6\) to \(6\) and \(t\) extends from \(0\) to \(\pi\).
- The sine wave starts at the origin, peaks at \(y = 6\) at \(t = \frac{\pi}{4}\), crosses back through zero at \(t = \frac{\pi}{2}\), reaches a minimum of \(y = -6\) at \(t = \frac{3\pi}{4}\), and returns to zero at \(t = \pi\).
2. **Second and Third Graphs:**
- Axis: \(y\) ranges from \(-1\) to \(1\) and \(t\) extends from \(0\) to \(\pi\).
- Similar to the scaled version, but reflects a standard sine wave for \(\sin(2t)\) without amplitude scaling. Peaks and troughs are at \(1\) and \(-1\) respectively.
Each sine graph follows the typical pattern of oscillation, illustrating one complete cycle over the interval from \(0\) to \(\pi\).
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