Laura's Way of Thinking About Linear and Angular Speed Problems

Really we are talking about three kinds of problems here that can all be done with one approach. I don't even worry about the formulas mentioned in this section. If you decide to do it my way, you won't need to worry about them either. The 3 kinds of things we are trying to find are two kinds of angular speed: rotations per unit of time and a unit of angle per unit of time and linear speed (a unit of distance per unit of time).

My approach is to just use unit analysis for all three. Keep in mind the units that your given information is in and the units that you want your answer in. Place the units for your answer on the right hand side of your equation and the units that your problem is given in on the left. Leave plenty of room for conversions.

Example: Suppose a bicyclist's is riding at a rate of 15 miles per hour. The diameter of his tires are 27 inches. There are 5280 feet in one mile.

1. How fast is a point on the circumference of the wheel spinning? (Give answer in inches per seconds.)
   
   

   Solution Start with the given: 15 mi/hr and work towards the goal of in/sec:

   (15mi/hr) × (1) × (1) × (1) × (1) = _____ in/sec
   then replace the 1's with fractions that are equivalent to 1, but where the units cancel with the given units and lead to the units we want in our answer. We want inches instead of miles so we convert miles to feet and then feet to inches. We want seconds instead of hours so we convert hours to minutes and minutes to seconds.
   

2. What is the angular speed of the tire?
   
   

   Solution Now we are looking for something like radians or degrees per second since the problem asks for angular speed. Just continue converting your answer to part a, keeping in mind that there are 2πr inches in one revolution and there are 2π radians in one revolution or you could just keep in mind from the definition of radians that there is one radius per radian, i.e. in this case since the radius is 13.5 in. we know that there are 13.5 inches per radian.

   

3. How many revolutions per minute are his tires making?
   
   

   Solution Let's start from scratch just for the heck of it. We start with 15 mi/hr and we want our answer to be in rev/min.