A bicycle has tyres of radius 0.35 meters. If the bicycle is traveling at a constant speed of `12 m//s`, at approximately what angular speed are the tyres rotating?

Options

A. `85 "rev"//"min"`
B. `197 "rev"//"min"`
C. `214 "rev"//"min"`
D. `327 "rev"//"min"`

Answer

To find the angular speed of the bicycle tyres, we can use the relationship between linear speed and angular speed. The formula connecting linear speed (v), angular speed (ω), and radius (r) is given by:

\[ v = r \cdot \omega \]

Where:
- \( v \) is the linear speed (in meters per second),
- \( r \) is the radius of the tyre (in meters),
- \( \omega \) is the angular speed (in radians per second).

### Step 1: Identify the given values
- Radius of the tyre, \( r = 0.35 \) meters
- Linear speed of the bicycle, \( v = 12 \) m/s

### Step 2: Rearrange the formula to solve for angular speed
We need to find \( \omega \), so we rearrange the formula:

\[ \omega = \frac{v}{r} \]

### Step 3: Substitute the values into the equation
Now substitute the known values into the equation:

\[ \omega = \frac{12 \, \text{m/s}}{0.35 \, \text{m}} \]

### Step 4: Calculate the angular speed
Perform the division:

\[ \omega = \frac{12}{0.35} \approx 34.29 \, \text{radians/second} \]

### Step 5: Convert angular speed to revolutions per minute (RPM)
To convert from radians per second to revolutions per minute, we use the conversion factor:

1 revolution = \( 2\pi \) radians

Thus, to convert radians per second to revolutions per minute:

\[ \text{RPM} = \omega \cdot \frac{1 \, \text{revolution}}{2\pi \, \text{radians}} \cdot \frac{60 \, \text{seconds}}{1 \, \text{minute}} \]

Substituting the value of \( \omega \):

\[ \text{RPM} = 34.29 \cdot \frac{1}{2\pi} \cdot 60 \]

### Step 6: Calculate the RPM
Now, calculate the value:

\[ \text{RPM} = 34.29 \cdot \frac{60}{2\pi} \approx 327.0 \, \text{RPM} \]

### Final Answer
The angular speed of the tyres is approximately **327 revolutions per minute (RPM)**.

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