An object in simple harmonic motion has a frequency of one sixth
oscillation per minute and an amplitude of 3 feet. Write an equation in the form d equals a sine omega t for the​ object's simple harmonic motion where t is in minutes.

To find the equation of the object's simple harmonic motion, we use the general form of a sine function for simple harmonic motion:

\[
d(t) = a \sin(\omega t)
\]

Where:
- \( d(t) \) is the displacement at time \( t \),
- \( a \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time in minutes.

### Given dataTo find the equation of the object's simple harmonic motion, we use the general form of a sine function for simple harmonic motion:

\[
d(t) = a \sin(\omega t)
\]

Where:
- \( d(t) \) is the displacement at time \( t \),
- \( a \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is time in minutes.

### Given 
- **Amplitude** \( a = 3 \) feet,
- **Frequency** = \( \frac{1}{6} \) oscillations per minute.

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

\[
\omega = 2 \pi f
\]

Substitute \( f = \frac{1}{6} \):

\[
\omega = 2 \pi \times \frac{1}{6} = \frac{\pi}{3} \, \text{radians per minute}.
\]

#### Step 2: Write the equation.
Now that we have the amplitude \( a = 3 \) feet and the angular frequency \( \omega = \frac{\pi}{3} \), the equation for the object's simple harmonic motion is:

\[
d(t) = 3 \sin\left(\frac{\pi}{3} t\right)
\]

This is the required equation for the object's motion.

Would you like further details on this process or have any questions?

Here are 5 related questions for exploration:
1. How would the equation change if the amplitude were doubled?
2. What happens to the motion if the frequency becomes 1 oscillation per minute?
3. How can we modify the equation if there is a phase shift in the motion?
4. What is the object's maximum velocity during its motion?
5. How do we calculate the period of this motion from the given equation?

**Tip:** Remember, the period of a simple harmonic motion is the inverse of the frequency, which helps in understanding the full cycle duration.

An object in simple harmonic motion has a frequency of one sixth oscillation per minute and an amplitude of 3 feet. Write an equation in the form d equals a sine omega t for the object's simple harmonic motion where t is in minutes.

Learn how to write the equation for an object in simple harmonic motion with an amplitude of 3 feet and a frequency of 1/6 oscillations per minute. This problem explores key concepts in trigonometry and simple harmonic motion, including the angular frequency formula.

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