What is this common angular speed? Answer in units of rad/s.

Question

The tape in a videotape cassette has a total length 255 m and can play for 1.8 h. As the tape starts to play, the full reel has an outer radius of 42 mm and an inner radius of 14 mm. At some point during the play, both reels will have the same angular speed. What is this common angular speed? Answer in units of rad/s.

Answer 1

For this question, you need to convert linear speed to angular speed. The only quantities you can use to get a linear speed is the length of the tape and the time it runs. Actually, this works well or else you may damage the tape or the cassette won't work properly otherwise. Now we can put the units in order and see what we get

The length is 255 m

and the time is 1.8 hours = 1.8 times 3600 s = 6480s so the linear velocity v= 255m/6480s =0/03935m/s.

The linear is related to the angular momentum by v=ωr

where ω is the angular velocity. So even though the linear velocity needs to stay constant, the angular velocity changes at both ends. At the beginning the radius value at one end is 42 mm and the radius value at the other end is 14 mm. When the two sides have the same angular speed they would need to have the same radius which is just the average of 42mm and 14 mm. You can verify that the average is 28 mm.

Thus the angular speed when both sides have the same speed (and radius) is ω =v/r =0.03935/28 =0.0014 rad/s