Question

question 25 (1 point)
a toy scooter has wheels with a diameter of 120 mm. what is the angular speed of
the wheels when the scooter is moving forward at 6.00 m/s?

a) 47.7 rpm

b) 72.0 rpm

c) 50.0 rpm

d) 100 rpm

e) none of the above.

Explanation:

Step1: Recall the relationship between linear speed (v) and angular speed (ω)

The formula relating linear speed \( v \), radius \( r \), and angular speed \( \omega \) (in rad/s) is \( v = r\omega \). First, we need to find the radius of the wheel. The diameter \( d = 120 \, \text{mm} = 0.12 \, \text{m} \), so the radius \( r=\frac{d}{2}=\frac{0.12}{2}=0.06 \, \text{m} \).

Step2: Calculate angular speed in rad/s

We know \( v = 6.00 \, \text{m/s} \) and \( r = 0.06 \, \text{m} \). From \( v = r\omega \), we can solve for \( \omega \): \( \omega=\frac{v}{r}=\frac{6.00}{0.06}=100 \, \text{rad/s} \).

Step3: Convert angular speed from rad/s to rpm

To convert from radians per second to revolutions per minute (rpm), we use the conversion factors: \( 1 \, \text{revolution} = 2\pi \, \text{radians} \) and \( 1 \, \text{minute} = 60 \, \text{seconds} \). So, \( \omega (\text{rpm})=\omega (\text{rad/s})\times\frac{60}{2\pi} \). Substituting \( \omega = 100 \, \text{rad/s} \), we get \( \omega=\frac{100\times60}{2\pi}=\frac{6000}{2\pi}=\frac{3000}{\pi}\approx954.93 \, \text{rpm} \)? Wait, no, wait, I made a mistake here. Wait, no, in step 2, I miscalculated \( \omega \). Wait, \( v = 6 \, \text{m/s} \), \( r = 0.06 \, \text{m} \), so \( \omega=\frac{v}{r}=\frac{6}{0.06}=100 \, \text{rad/s} \)? No, that's wrong. Wait, 6 divided by 0.06 is 100? Wait, 0.06 times 100 is 6, yes. But then converting to rpm: \( \omega (\text{rpm})=\frac{\omega (\text{rad/s})\times60}{2\pi} \). So \( 100 \times\frac{60}{2\pi}=\frac{6000}{2\pi}=\frac{3000}{\pi}\approx954.93 \, \text{rpm} \). But this is not matching the options. Wait, maybe I messed up the radius conversion. Wait, diameter is 120 mm, which is 0.12 meters, so radius is 0.06 meters, that's correct. Linear speed is 6 m/s. Wait, maybe the question has a typo? Or maybe my approach is wrong? Wait, no, wait, maybe I confused the formula. Wait, another way: the distance covered in one revolution is the circumference \( C = \pi d= \pi\times0.12 \, \text{m} \). The number of revolutions per second (frequency \( f \)) is \( f=\frac{v}{C}=\frac{6}{\pi\times0.12} \). Then, revolutions per minute (rpm) is \( f\times60 \). Let's calculate that. \( f=\frac{6}{0.12\pi}=\frac{50}{\pi} \, \text{rev/s} \). Then rpm is \( \frac{50}{\pi}\times60=\frac{3000}{\pi}\approx954.93 \, \text{rpm} \). But the options are 47.7, 72, 50, 100, or none. So the correct answer should be e) None of the above. Wait, but maybe I made a mistake in units. Wait, the diameter is 120 mm, which is 12 cm, 0.12 m. Linear speed is 6 m/s. Let's recalculate:

Circumference \( C = \pi d = \pi\times0.12 \, \text{m} \approx 0.37699 \, \text{m} \) per revolution.

Number of revolutions per second: \( n=\frac{v}{C}=\frac{6}{0.37699}\approx15.915 \, \text{rev/s} \)

Number of revolutions per minute: \( 15.915\times60\approx954.9 \, \text{rpm} \), which is not among the options a - d. So the answer is e.

Wait, but maybe I messed up the radius. Wait, diameter 120 mm is 0.12 m, radius 0.06 m. Linear speed v = rω, so ω = v/r = 6 / 0.06 = 100 rad/s. Convert to rpm: 100 rad/s (60 s / 1 min) (1 rev / 2π rad) = (100*60)/(2π) = 6000/(2π) = 3000/π ≈ 954.9 rpm. So none of the options a - d match, so the answer is e.

Answer:

e) None of the above.