Question

Write a sine or cosine function for 70 revolutions per minute and an amplitude of 12

Answer 1

To write a sine or cosine function for a sinusoidal motion with a frequency of 70 revolutions per minute (RPM) and an amplitude of 12, we need to convert the RPM to radians per second.

### Step 1: Convert RPM to Radians Per Second
1 revolution corresponds to \(2\pi\) radians. Therefore, to convert RPM to radians per second, we use the conversion factor:

\[
\text{Radians per second} = \text{RPM} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}}
\]

For 70 RPM:
\[
\text{Frequency in radians/second} = 70 \times \frac{2\pi}{60} = \frac{140\pi}{60} = \frac{14\pi}{6} = \frac{7\pi}{3} \text{ radians per second}
\]

### Step 2: Write the Sine or Cosine Function
The general form of a sine or cosine function is:
- For cosine:
\[
y(t) = A \cos(\omega t + \phi)
\]
- For sine:
\[
y(t) = A \sin(\omega t + \phi)
\]

Where:
- \(A\) is the amplitude (which is 12),
- \(\omega\) is the angular frequency (which is \(\frac{7\pi}{3}\)),
- \(t\) is time,
- \(\phi\) is the phase shift (which you can set to 0 if there's no phase shift).

Choose cosine for this example:

### Final Function
\[
y(t) = 12 \cos\left(\frac{7\pi}{3} t\right)
\]

Alternatively, if you prefer to use the sine function for the same parameters, it could be:

\[
y(t) = 12 \sin\left(\frac{7\pi}{3} t\right)
\]

Both functions represent the same motion with the specified amplitude and frequency.