Question

A certain merry go round has a radius of 5 meters and an angular velocity of 1/4 pi radians per second. A) how long does it take the merry go round to make one complete revolution? B) How many turns does it make in one minute?

I need the equations as well as an explanation

Answer 1

You're given an angular velocity in \[\left[ \frac{ rads }{ \sec } \right]\] which is a measurement of the travel over an angle of our circle. Intuitively you can decide that this makes the radius of our circle (5m) irrelevant, since the angle is constant no matter how large or small our circle is. If you were asked for, say, distance traveled then the radius would matter because of the circumference of the circle changing.

When dealing in the unit of radians, 2pi equals 360 degrees, or one complete revolution. With this we can work towards the solution to part "A)".

A) \[2\pi [rads] = \frac{ \pi }{ 4 }\left[ \frac{ rads }{ \sec } \right] t [\sec] = 8 [\sec]\]This equation relates the total angular travel to the angular travel per second. It's the same as velocity = distance over time but in terms of circle instead of a line. Hopefully I've helped make that clear. The units are in brackets. The equation can be solved without them there but it is good practice to carry them. If you cancel them correctly your answer should always make sense.

B) This is a much simpler extension. If it takes you 8 seconds to make one turn then how many turns will you make in 60 seconds? Immediately you can say that it will be 60/8 turns and you can prove this by setting up your equation, again being conscious of your units.\[\frac{ 1 }{ 8 }\left[ \frac{ turn }{ \sec } \right]60\left[ \frac{ \sec }{ \min } \right] = \frac{ 60 }{ 8 }\left[ \frac{ turn }{ \min } \right]\]