Question

a father is helping his young son learn to ride a bike by pushing on the bike seat from the back. at one point, the wheel of the bike is turning with an angular speed of one revolution in 6 seconds. later, after 18.1 seconds have elapsed, it is turning with an angular speed of 26 revolutions per minute. what angle did the wheel turn over these 18.1 seconds? hint: first, find out the average angular acceleration of the bicycle wheel in rad/sec² during this time interval. your answer: 34.11 answer units

Explanation:

Step1: Convert initial angular speed to rad/sec

The initial angular speed $\omega_0$ is 1 revolution in 6 seconds. Since 1 revolution = $2\pi$ radians, $\omega_0=\frac{2\pi}{6}=\frac{\pi}{3}$ rad/sec.

Step2: Convert final angular speed to rad/sec

The final angular speed $\omega$ is 26 revolutions per minute. First convert to revolutions per second: $\frac{26}{60}$ revolutions per second. Then convert to rad/sec: $\omega=\frac{26}{60}\times2\pi=\frac{13\pi}{15}$ rad/sec.

Step3: Calculate average angular acceleration

The formula for average angular acceleration $\alpha$ is $\alpha=\frac{\omega - \omega_0}{t}$, where $t = 18.1$ seconds. So $\alpha=\frac{\frac{13\pi}{15}-\frac{\pi}{3}}{18.1}=\frac{\frac{13\pi - 5\pi}{15}}{18.1}=\frac{\frac{8\pi}{15}}{18.1}=\frac{8\pi}{15\times18.1}$ rad/sec².

Step4: Use angular - displacement formula

The angular - displacement formula is $\theta=\omega_0t+\frac{1}{2}\alpha t^{2}$. Substitute $\omega_0=\frac{\pi}{3}$, $\alpha=\frac{8\pi}{15\times18.1}$, and $t = 18.1$ into the formula:
\[

$$\begin{align*} \theta&=\frac{\pi}{3}\times18.1+\frac{1}{2}\times\frac{8\pi}{15\times18.1}\times(18.1)^{2}\\ &=\frac{18.1\pi}{3}+\frac{4\pi\times18.1}{15}\\ &=\frac{90.5\pi + 24.133\pi}{15}\\ &=\frac{114.633\pi}{15}\\ &\approx 23.9 \end{align*}$$

\]
There is a mistake above. We can also use the formula $\theta=\frac{\omega_0+\omega}{2}\times t$.
Substitute $\omega_0=\frac{\pi}{3}$ rad/sec and $\omega=\frac{13\pi}{15}$ rad/sec and $t = 18.1$ seconds.
\[

$$\begin{align*} \theta&=\frac{\frac{\pi}{3}+\frac{13\pi}{15}}{2}\times18.1\\ &=\frac{\frac{5\pi + 13\pi}{15}}{2}\times18.1\\ &=\frac{\frac{18\pi}{15}}{2}\times18.1\\ &=\frac{3\pi}{5}\times18.1\\ &=\frac{54.3\pi}{5}\\ &\approx 34.11 \end{align*}$$

\]

Answer:

34.11 radians