The position of a particle in cm is given by x = (8) cos 6nt, where t is in seconds. (a) What is the frequency? Hz (b) What is the period? (c) What is the amplitude of the particle's motion? cm (d) What is the first time after t = 0 that the particle is at its equilibrium position? In what direction is it moving at that time? O in the positive direction in the negative direction
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.
![**Harmonic Motion Analysis**
The position of a particle in cm is given by the equation \( x = (8) \cos{6\pi t} \), where \( t \) is in seconds.
### Questions:
**(a)** What is the frequency?
\[ \text{Frequency:} \_\_\_\_ \text{ Hz} \]
**(b)** What is the period?
\[ \text{Period:} \_\_\_\_ \text{ s} \]
**(c)** What is the amplitude of the particle's motion?
\[ \text{Amplitude:} \_\_\_\_ \text{ cm} \]
**(d)** What is the first time after \( t = 0 \) that the particle is at its equilibrium position?
\[ \text{Time:} \_\_\_\_ \text{ s} \]
#### In what direction is it moving at that time?
- \( \bigcirc \) in the positive direction
- \( \bigcirc \) in the negative direction
### Explanation of Terms:
1. **Frequency (f)**: The number of oscillations per second.
2. **Period (T)**: The time taken for one complete cycle of oscillation.
3. **Amplitude (A)**: The maximum displacement from the equilibrium position.
4. **Equilibrium position**: The position where the net force on the particle is zero, typically at \( x = 0 \).
### Steps to Solve:
**(a) Calculating Frequency:**
\[ x = 8 \cos{6\pi t} \]
The term inside the cosine function \((6\pi t)\) has the form \( \omega t \), where \( \omega \) is the angular frequency. Here,
\[ \omega = 6\pi \]
The relationship between angular frequency (\( \omega \)) and frequency (f) is:
\[ \omega = 2\pi f \]
Hence,
\[ 6\pi = 2\pi f \]
\[ f = \frac{6\pi}{2\pi} \]
\[ f = 3 \text{ Hz} \]
**(b) Calculating Period:**
The period \( T \) is the reciprocal of the frequency (\( f \)).
\[ T = \frac{1}{f} \]
Given \( f = 3 \text{ Hz} \):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00822f5e-0c6f-4029-81e1-6a6ab50c835f%2Fe6c77f7f-b24b-485d-b1e1-d26f6cc7c865%2F4r2s64u.jpeg&w=3840&q=75)
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