To solve this problem, we need to find the angular acceleration of the bicycle wheel. Angular acceleration ([tex]\(\alpha\)[/tex]) is defined as the rate of change of angular velocity ([tex]\(\omega\)[/tex]) over time. We can follow these steps to find the solution:
1. Convert the initial and final linear velocities to angular velocities using the "pirate equation" [tex]\(v = r \omega\)[/tex], where [tex]\(v\)[/tex] is linear velocity and [tex]\(r\)[/tex] is the radius of the wheel.
2. Calculate the initial angular velocity ([tex]\(\omega_i\)[/tex]) and the final angular velocity ([tex]\(\omega_f\)[/tex]).
3. Determine the angular acceleration ([tex]\(\alpha\)[/tex]) using the definition: [tex]\(\alpha = \frac{\Delta \omega}{\Delta t}\)[/tex].
### Step-by-Step Solution:
1. Given values:
- Radius of the wheel, [tex]\(r = 0.29\)[/tex] meters
- Initial linear velocity, [tex]\(v_i = 0\)[/tex] m/s
- Final linear velocity, [tex]\(v_f = 5\)[/tex] m/s
- Time interval, [tex]\(\Delta t = 2.6\)[/tex] seconds
2. Convert linear velocities to angular velocities:
Using the relationship [tex]\(v = r \omega\)[/tex], we can find the angular velocities.
- Initial angular velocity:
[tex]\[
\omega_i = \frac{v_i}{r} = \frac{0}{0.29} = 0 \, \text{rad/s}
\][/tex]
- Final angular velocity:
[tex]\[
\omega_f = \frac{v_f}{r} = \frac{5}{0.29} \approx 17.24 \, \text{rad/s}
\][/tex]
3. Calculate the angular acceleration [tex]\(\alpha\)[/tex]:
Angular acceleration is given by:
[tex]\[
\alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega_f - \omega_i}{\Delta t}
\][/tex]
Substituting the values we found:
[tex]\[
\alpha = \frac{17.24 \, \text{rad/s} - 0 \, \text{rad/s}}{2.6 \, \text{s}} \approx 6.63 \, \text{rad/s}^2
\][/tex]
4. Conclusion:
The angular acceleration of the bicycle wheel is approximately [tex]\(6.63 \, \text{rad/s}^2\)[/tex].
