An object is moving back and forth on the x-axis according to the equation x(t) = 3sin(20πt), t> 0, where x(t) is measured in cm and t in seconds, How many complete back-and-forth motions (from the origin to the right, back to the origin, to the left and finally back to the origin) does the object make in one second

The given motion equation is:

\[
x(t) = 3 \sin(20\pi t)
\]

This represents a sinusoidal oscillation, where \( x(t) \) is the position of the object along the x-axis in centimeters, and \( t \) is time in seconds. The general form for sinusoidal motion is:

\[
x(t) = A \sin(\omega t)
\]

where:
- \( A \) is the amplitude of oscillation (in this case, \( A = 3 \) cm),
- \( \omega \) is the angular frequency (in this case, \( \omega = 20\pi \)).

### Step 1: Calculate the frequency of oscillation
The angular frequency \( \omega \) is related to the frequency \( f \) by the formula:

\[
\omega = 2\pi f
\]

Substituting \( \omega = 20\pi \):

\[
20\pi = 2\pi f
\]

Solving for \( f \):

\[
f = \frac{20\pi}{2\pi} = 10 \, \text{Hz}
\]

This means the object completes 10 oscillations per second.

### Step 2: Back-and-forth motion
One complete back-and-forth motion (from the origin, to the right, back to the origin, to the left, and back to the origin) corresponds to one full cycle of the sine function. Therefore, the object makes 10 complete back-and-forth motions in one second.

### Final Answer:
The object makes **10 complete back-and-forth motions** in one second.

Would you like further details or have any questions?

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**Relative Questions:**
1. How would the amplitude affect the motion in this case?
2. What is the period of the object's motion?
3. How would the motion change if the angular frequency were halved?
4. What is the maximum velocity of the object?
5. How would the motion change if the equation were \( x(t) = 3 \cos(20\pi t) \)?

**Tip:** The frequency of a sinusoidal motion can be easily determined from the angular frequency using the formula \( \omega = 2\pi f \).

An object is moving back and forth on the x-axis according to the equation x(t) = 3sin(20πt), t> 0, where x(t) is measured in cm and t in seconds, How many complete back-and-forth motions (from the origin to the right, back to the origin, to the left and finally back to the origin) does the object make in one second?

Learn how to calculate the number of complete back-and-forth motions an object makes on the x-axis using the equation x(t) = 3sin(20πt). This detailed solution covers concepts of oscillatory motion, angular frequency, and sinusoidal motion, suitable for Grades 11-12.

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