A bicyclist speeds up from 4.25 m/s to 6.75 m/s in 1.25 seconds. Given that the bicycle wheels have a radius of 31.1 cm, what angular acceleration did they have assuming it was constant?

To find the angular acceleration (\(\alpha\)) of the bicycle wheels, we’ll first calculate the linear acceleration of the bicyclist and then relate it to angular acceleration using the radius of the wheels.

### Step 1: Calculate Linear Acceleration
The linear acceleration (\(a\)) can be found using the formula:
\[
a = \frac{v_f - v_i}{t}
\]
where:
- \(v_f = 6.75 \, \text{m/s}\) (final velocity),
- \(v_i = 4.25 \, \text{m/s}\) (initial velocity),
- \(t = 1.25 \, \text{s}\) (time taken).

Substituting the values:
\[
a = \frac{6.75 - 4.25}{1.25} = \frac{2.5}{1.25} = 2.0 \, \text{m/s}^2
\]

### Step 2: Relate Linear Acceleration to Angular Acceleration
The relationship between linear acceleration (\(a\)) and angular acceleration (\(\alpha\)) is given by:
\[
a = r \alpha
\]
where \(r\) is the radius of the wheel.

Rearrange to solve for \(\alpha\):
\[
\alpha = \frac{a}{r}
\]

### Step 3: Substitute the Radius
Convert the radius from centimeters to meters:
\[
r = 31.1 \, \text{cm} = 0.311 \, \text{m}
\]

Now, substitute \(a = 2.0 \, \text{m/s}^2\) and \(r = 0.311 \, \text{m}\):
\[
\alpha = \frac{2.0}{0.311} \approx 6.43 \, \text{rad/s}^2
\]

### Answer
The angular acceleration of the bicycle wheels is approximately:
\[
\alpha \approx 6.43 \, \text{rad/s}^2
\]

Would you like further details or have any questions?

---

1. What is angular velocity and how does it relate to linear velocity?
2. How would the problem change if the radius of the wheel were different?
3. What would happen to angular acceleration if the bicyclist accelerated over a longer time?
4. How is angular acceleration useful in understanding rotational motion?
5. Could we determine the number of wheel rotations from this data?

**Tip:** Remember that linear and angular quantities can be related by the radius in circular motion, making it easy to convert between the two.

Discover how to find the angular acceleration of bicycle wheels when a bicyclist speeds up from 4.25 m/s to 6.75 m/s in 1.25 seconds. This problem involves concepts of linear and angular acceleration and uses a wheel radius of 31.1 cm, making it perfect for high school physics students.

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