Velocity, Radius and Angular Speed - Finding Units

I have been trying to figure this one out for two days, and I really have no idea what to do.

To begin with, I need to know what units will be produced in the following situations:

• Velocity's units when angular speed and radius are rpm and feet, respectively. (I think it is feet/minute)
• Anuglar speed's units when radius and velocity are foot and feet per minute, respectively. (It stays as feet/minute?)
• Radius' units when velocity and angular speed are mph and rpm, respectively. (I am guessing it will come out as miles, according to what my teacher tried to communicate)

Those are just a few issues, but there are more. I believe that, if I am correct/find why I am incorrect, I will be able to apply the concept to the other problems.

The second problem I am having is that I have a problem where: Velocity is 50 feet/second, Angular Speed is 100 revolutions/second and I need to find the radius in miles. According to what I know, velocity is angular speed times theta, so I just have to divide by angular speed. However, I do not know where to go when there are different units.

I know it is a lot of text, but I appreciate anyone who can help me; my teacher said she is preoccupied and cannot answer any more of my questions today, so I'm stuck.

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2 Answers

I always think of this sort of stuff in terms of forcing the the result to have the units you want and/or the only units that can possibly make sense.

In the first case, you are given $\frac{rotations}{minutes}$ and $feet=\frac{feet}{1}$ and you want to end up with velocity, which has the general form $\frac{distance}{time}$. In this case, the pertinent units for distance and time are feet and minutes, respectively. The key, though, is the following: one rotation equals $2\pi$ times the radius (in feet). That is, $$\frac{rotations}{minutes}=2\pi\cdot radius\cdot\frac{feet}{minutes}.$$

But remember that in the end we just want an expression for velocity; that is, $\frac{feet}{minutes}=$ something. That's simple, though. Just divide both sides of the above equation by $2\pi\cdot radius$.

The second and third ones are very similar, but angular velocity has units $\frac{rotations}{minutes}$.

The last thing you asked is a specific instance of this sort of dimensional analysis, except that you have to convert everything to miles in the end. Alternatively, you can think of it as follows: if the velocity is 50 feet/second and the angular speed is 100 revolutions/second, that means in one second, the apparatus is making 100 revolutions - and furthermore, that these 100 revolutions = 50 feet. From this you get that one revolution is 1/2 a foot. That means that the circumference of the circle drawn out by the motion of the apparatus is 1/2 a foot, meaning that the radius of the circle is $\frac{1}{2}\cdot\frac{1}{2\pi}$ feet. To turn this into miles, just divide by 5280. Hope that helps.

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Yay! Much obliged. – nmagerko Feb 22 '12 at 23:52

Angular speed has to be some measure of angle divided by some measure of time. In your second bullet point it will probably be revolutions per minute (rpm).

You presumably have some formula saying velocity is related the product of radius and angular speed. Here you probably have $$50 \text{ feet/ second} = r \times 2\pi \text{ radians/revolution }\times 100 \text{ revolutions/ second}$$ which gives a radius of $r = 0.25 / \pi \text{ feet}$, not very much. If you must convert it into miles, then use $1 \text{ mile} = 5280 \text{ feet}$, though that will give you an even smaller number.

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That is why I don't understand why we do problems like this! – nmagerko Feb 23 '12 at 0:42
It does get slightly easier using SI units. – Henry Feb 23 '12 at 0:54