To solve this, we need to find the angular speed in radians per minute and the number of revolutions per minute.

Step 1: Find the angular speed in rad/min

The angular speed $\omega$ can be found using the relationship between linear speed and the radius of the wheel. We need to convert the bicycle’s linear speed into the angular speed.

The linear speed (in miles per hour, $v = 17 \, \text{mi/h}$) is related to the angular speed $\omega$ by the following formula:

$v = r \cdot \omega$

where:

• $v$ is the linear speed,
• $r$ is the radius of the wheel, and
• $\omega$ is the angular speed in radians per minute.

We need to convert the linear speed into inches per minute:

• $1 \, \text{mile} = 5280 \, \text{feet}$,
• $1 \, \text{foot} = 12 \, \text{inches}$,
• $1 \, \text{hour} = 60 \, \text{minutes}$.

So, we first convert $17 \, \text{mi/h}$ to inches per minute.

$v = 17 \, \text{mi/h} \times \frac{5280 \, \text{ft}}{1 \, \text{mile}} \times \frac{12 \, \text{inches}}{1 \, \text{ft}} \times \frac{1 \, \text{hour}}{60 \, \text{minutes}}$

Now, let's calculate this:

$v = 17 \times 5280 \times 12 \div 60$

After simplifying:

$v = 17 \times 1056 = 17952 \, \text{inches/min}$

Now that we know the linear speed is $17952 \, \text{inches/min}$, we need the radius $r$. The diameter of the wheel is 20 inches, so the radius $r = 10 \, \text{inches}$.

Now we can solve for the angular speed $\omega$:

$\omega = \frac{v}{r} = \frac{17952}{10} = 1795.2 \, \text{rad/min}$

Step 2: Find the number of revolutions per minute

The number of revolutions per minute (RPM) is the number of times the wheel completes a full rotation in one minute. One revolution corresponds to $2\pi$ radians. So, the number of revolutions per minute is:

$\text{RPM} = \frac{\omega}{2\pi} = \frac{1795.2}{2\pi}$

Using $\pi \approx 3.1416$, we get:

$\text{RPM} = \frac{1795.2}{6.2832} \approx 285.451$

Final Answers:

• The angular speed $\omega = 1795.2 \, \text{rad/min}$.
• The number of revolutions per minute (RPM) is approximately 285.451.

Let me know if you'd like further explanation on any of these steps!

Related Questions:

1. How does the radius of the wheel affect the angular speed?
2. How can we calculate angular speed in radians per second instead of radians per minute?
3. What would happen if the speed of the bicycle were increased?
4. How do we calculate the linear velocity of a rotating object given its angular speed and radius?
5. Why is it important to use consistent units when calculating angular velocity?

Tip: To make conversions easier, always ensure that units cancel out properly, and remember to convert both time and distance into compatible units (e.g., minutes and inches, in this case).