Circular motion and linear speed of an object

[ragbash]

An object is traveling around a circle with the radius of 5cm. If in 20sec the central angle of 1/3 radian is swept out, what is the angular speed of the object? Linear speed?

Here's how I did it. angular speed-->
a) omega=theta(in radians)/elapsed time
= π/3/20= π/3*1/20 = π/60 radians/sec. Is that the same as the answer in the book, 1/60?

linear speed-->
b) v=rw (length/radius)(omega=angular speed)
=5cm*(1/60)= 1/12 cm/s
the answer in the book is 12m/s.

what did I do wrong?

 

[interested_learner]

Where did you get the n from? You went 1/3 radian in 20 seconds. To get the radian speed just find the distance traveled in one second. The book is right.

If it really was 12 m/s it would have gone around the circle about 60 times in 20 seconds since the length of the circumference is 2 * pi * radius or .314 m. So the book is wrong. You are wrong too. Remember a radian is radius / arch distance. So r/s = 1/3 in 20 seconds. Once you have arch distance traveled in 20 seconds, finding linear speed is easy.

 

[HallsofIvy]

interested learner, I don't think it was "n", it was $\pi$ using an overly simple font.

ragbash, your problem says "1/3 radian". For some reason you used "$\pi/3$ radians.

No, $\pi/3$ is not the same as 1/3!

The circumference of a circle is $2\pi r$ or, since r= 5 here, $10\pi$ cm. Since the object moves 1/60 radian/sec and there are $2\pi$radians in a circle, it is moving at $\frac{1}{120\pi}$ "circles per second" and so [itex]\frac{1}{120\pi}(10\pi)= 1/12$cm/sec.<br /> Your book apparently has a typo for the second.$[/itex]