Question

question 6
a truck with 32 - inch diameter wheels is traveling at 50 mi/h. round answers to the nearest whole number.
find the angular speed of the wheels in rad/min.
rad/min
how many revolutions per minute do the wheels make?
rev/min
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Explanation:

Step1: Convert linear speed to inches per minute

We know that \(1\) mile \( = 5280\) feet and \(1\) foot \(=12\) inches. So \(1\) mile \(=5280\times12 = 63360\) inches. And \(1\) hour \( = 60\) minutes.
The linear speed \(v = 50\) mi/h. Converting to inches per minute:
\(v=50\times\frac{63360}{60}\) inches/min.
\(v = 52800\) inches/min.

Step2: Find the radius of the wheel

Given diameter \(d = 32\) inches, radius \(r=\frac{d}{2}=16\) inches.

Step3: Use the formula \(v = r\omega\) to find angular speed \(\omega\)

The formula relating linear speed \(v\), radius \(r\), and angular speed \(\omega\) is \(v=r\omega\).
We solve for \(\omega\): \(\omega=\frac{v}{r}\).
Substituting \(v = 52800\) inches/min and \(r = 16\) inches.
\(\omega=\frac{52800}{16}=3300\) rad/min.

Step4: Convert angular speed to revolutions per minute

We know that \(1\) revolution \(=2\pi\) radians.
Let \(n\) be the number of revolutions per minute. Then \(\omega = 2\pi n\).
\(n=\frac{\omega}{2\pi}\).
Substituting \(\omega = 3300\) rad/min.
\(n=\frac{3300}{2\pi}\approx\frac{3300}{2\times3.14}\approx 525\) rev/min.

Answer:

The angular speed of the wheels is \(3300\) rad/min.
The number of revolutions per minute is \(525\) rev/min.