A truck with 46-in.-diameter wheels is traveling at 50 mi/h. Find the angular speed of the wheels in radians

Question

A truck with 46-in.-diameter wheels is traveling at 50 mi/h. Find the angular speed of the wheels in radians

Answer 1

Hi Chole.

I'll be happy to help you with this problem, but I think you left something out.

Angular speed has to be expressed in radian per (some unit of time).

No worries I'll show you how to work which ever ones you need later.

Essentially, we need to convert a linear speed measured in miles per hour to an angular speed measured in radians per (some unit of time)

First, we'll need to calculate the circumference of the wheels.

Since the diameter is 46 inches, the circumference is 46pi inches.

This means that each time the wheel rotates it move forward by 46pi inches.

Now I'm going to use dimensional analysis to make this foolproof.

we're going to convert 50 miles/hour to some number of radians/hour.

50 miles/hour x (5280 feet/1 mile) x (12 inches/1 foot) x (1 rotation/46pi inches) x (2pi radians/1 rotation)

Notice each time a unit appears in the numerator of one fraction and the denominator of another THEY CANCEL OUT!

miles cancel, feet cancel, inches cancel, rotations cancel. (Notice even the pi's cancel, too.)

What's left? Radians per hour.

Now we multiply the 50 by every number that appears in a numerator and divide by all the number that appear in the denominators. (In this case it's only the 46.)

Now if we multiply our answer X radians/hour x (1 hour/60 minutes) and we'll have an answer in radians/minute.

Multiply THAT by (1 minute/60 seconds) and our answer will be in radians/second.

You gotta love dimensional analysis. It makes the math make sense.

I hope this works.

Answer 2

Hi Chloe.

This question requires a ton of unit changes. Angular speed is measured in radians/ sec. The wheel's diameter is measured in inches. The speed of the truck is giving in miles per hour.

Let's start by converting 50 miles per hour to inches per second.

50 miles / hour * 5280 feet / mile * 12 inches / foot * 1 hour / 60 minutes * 1 minute / 60 seconds

= (50 * 5280 * 12) / (60 * 60) = 880 inches / sec

Your truck is traveling 880 inches per second.

Your truck wheel has a diameter of 46 inches. This translates to a circumference of 46π inches. Every time the truck wheel rotates, it travels a horizontal distance of 46π inches.

We know the horizontal speed is 880 inches per second, so that means:

880 inches / sec * 1 rotation / 46π inches = 880/46π rotations/ sec

The question wants the answer in radians per second rather than rotations per second. We know one full rotation of a circle is equal to 2π radians, so there's just one more conversion to make.

880/46π rotations/ sec * 2π radians / rotation

= 880π / 23π radians / sec

= ~38.26 radians/ sec