Angular velocity error in textbook?

[lonewolf219]

The text for the intro physics class I'm taking is the first edition. I have noticed some errors in earlier chapters, and was wondering if this problem here is also an error? Not sure yet.

The question is about a CD. It says to find the angular velocity, and gives two points on the radius. r_1 and r_2. The question is an example, and the problem is solved for you. The given tangential velocity is 1.216 m/s. Then, they divide v_t by each radius, and determine two different angular velocities.

Is this correct? My understanding is only the tangential velocity can change depending upon position on the radius. Does this mean the angular velocity also depends on the radius?

ω_1 is 48 rad/s
ω_2 is 20 rad/s

 

[grzz]

When a CD is rotating about its axis, all points on it have the SAME angular speed but the tangential speed at a point on the CD depends on the distance of given point from the centre.

I think that I overlooked how a CD really works.

 

[lonewolf219]

Ok, so there should only be one angular speed. Was the mistake made by using the same tangential velocity for both r_1 and r_2?

 

[grzz]

Ok, so there should only be one angular speed. Was the mistake made by using the same tangential velocity for both r_1 and r_2?

One would require values of r1 and r2 to check that.

 

[lonewolf219]

v_t= 1.216 m/s
r_1= 0.025 m
r_2= 0.058 m
ω_1= 48.64 rad/s
ω_2= 20.97 rad/s

Here are the values they used

 

[grzz]

v_t= 1.216 m/s
r_1= 0.025 m
r_2= 0.058 m
ω_1= 48.64 rad/s
ω_2= 20.97 rad/s

v = r\omega

Why don't you test it whether the tangential speed is being mixed up with the angular speed?

 

[lonewolf219]

OK. Thanks for your help, Grzz. Glad physics forums is so accessible

 

[AlephZero]

Maybe the question didn't explain clearly how the numbers relate to a real audio CD.

The data on the CD is recorded at a constant linear (tangential) velocity of about 1.2 m/s over the whole disk. The data is recorded on a spiral track, starting at the smallest radius and working outwards.

As the radius changes, the angular velocity of the CD player slows down as it plays the disk from the inside to the outside. The radii of 25mm and 58mm in your question are the inner and outer radius of the recording area. The question is asking you to fnd the max and min speeds of the disk.

Note, the question is talking about playing an audio CD in "real time". If you are ripping the contents using a computer CD drive, the angular velocity might be 40 or 50 times faster.

 

[lonewolf219]

Wow. Gotcha. Great information, AlephZero, thanks!