Question
Write a sine or cosine function for a diameter of 24 inches and 70 revolutions per minute.
Answers
GPT-4o mini
To create a sine or cosine function representing a circular motion with a diameter of 24 inches and a speed of 70 revolutions per minute (RPM), we can use the general form:
\[ y(t) = A \cdot \sin(Bt + C) + D \]
or
\[ y(t) = A \cdot \cos(Bt + C) + D \]
### Step 1: Determine the parameters
1. **Amplitude (A)**: This is half of the diameter, so:
\[
A = \frac{24}{2} = 12 \text{ inches}
\]
2. **Vertical Shift (D)**: Since we want the motion to oscillate around a central line, we can set this to 0 for simplicity:
\[
D = 0
\]
3. **Angular frequency (B)**: To find this, we need to convert revolutions per minute to radians per second because sine and cosine functions use radians.
- One full revolution is \(2\pi\) radians.
- 70 revolutions per minute can be converted to radians per second as follows:
\[
70 \text{ rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ sec}} = \frac{70 \times 2\pi}{60} = \frac{7\pi}{6} \text{ radians/sec}
\]
Thus, we have:
\[
B = \frac{7\pi}{6}
\]
4. **Phase shift (C)**: If we start at the maximum displacement (the top position), we can set the cosine function without a phase shift:
\[
C = 0
\]
### Step 2: Write the function
Putting it all together, we can express the circular motion in either the sine or cosine form. Using the cosine function, the equation will be:
\[
y(t) = 12 \cdot \cos\left(\frac{7\pi}{6} t\right)
\]
Alternatively, you could use the sine function:
\[
y(t) = 12 \cdot \sin\left(\frac{7\pi}{6} t + \frac{\pi}{2}\right)
\]
### Final Form
You can use either of the following equations to represent the motion of the object with diameter 24 inches rotating at 70 RPM:
1. Cosine function:
\[
y(t) = 12 \cdot \cos\left(\frac{7\pi}{6} t\right)
\]
2. Sine function:
\[
y(t) = 12 \cdot \sin\left(\frac{7\pi}{6} t + \frac{\pi}{2}\right)
\]
You can select either equation based on the initial position of the object in its motion.
\[ y(t) = A \cdot \sin(Bt + C) + D \]
or
\[ y(t) = A \cdot \cos(Bt + C) + D \]
### Step 1: Determine the parameters
1. **Amplitude (A)**: This is half of the diameter, so:
\[
A = \frac{24}{2} = 12 \text{ inches}
\]
2. **Vertical Shift (D)**: Since we want the motion to oscillate around a central line, we can set this to 0 for simplicity:
\[
D = 0
\]
3. **Angular frequency (B)**: To find this, we need to convert revolutions per minute to radians per second because sine and cosine functions use radians.
- One full revolution is \(2\pi\) radians.
- 70 revolutions per minute can be converted to radians per second as follows:
\[
70 \text{ rev/min} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ sec}} = \frac{70 \times 2\pi}{60} = \frac{7\pi}{6} \text{ radians/sec}
\]
Thus, we have:
\[
B = \frac{7\pi}{6}
\]
4. **Phase shift (C)**: If we start at the maximum displacement (the top position), we can set the cosine function without a phase shift:
\[
C = 0
\]
### Step 2: Write the function
Putting it all together, we can express the circular motion in either the sine or cosine form. Using the cosine function, the equation will be:
\[
y(t) = 12 \cdot \cos\left(\frac{7\pi}{6} t\right)
\]
Alternatively, you could use the sine function:
\[
y(t) = 12 \cdot \sin\left(\frac{7\pi}{6} t + \frac{\pi}{2}\right)
\]
### Final Form
You can use either of the following equations to represent the motion of the object with diameter 24 inches rotating at 70 RPM:
1. Cosine function:
\[
y(t) = 12 \cdot \cos\left(\frac{7\pi}{6} t\right)
\]
2. Sine function:
\[
y(t) = 12 \cdot \sin\left(\frac{7\pi}{6} t + \frac{\pi}{2}\right)
\]
You can select either equation based on the initial position of the object in its motion.