Question

The tape in a videotape cassette has a total length 187 m and can play for 1.7 h. As the tape starts to play, the full reel has an outer radius of 36 mm and an inner radius of 12 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

Answer:

0.114 rad/s

Explanation:

As the plastic tape is coming off from 1 reel onto another reel, the velocity of the 2 reels must be the same. For the angular speed to be the same, their radius must be the same as well since:

$v = \omega * r$

Let this radius be r, we know that 12 < r < 36 mm and r is somewhere so that the area of the donut-shaped object between 12mm and r is the same as the area of the donut-shaped object between r and 36mm

In math terms:

$\pi r^2 – \pi 12^2 = \pi 36^2 – \pi r^2$

From here we can divide both sides by  π

$r^2 - 144 = 1296 - r^2$

$2r^2 = 1440$

$r^2 = 720$

$r = \sqrt{720} = 26.8mm$ or 0.0268m

1.7 hours = 1.7*3600 = 6120 seconds

The velocity of the tapes is 187 / 6120 = 0.031 m/s

And so the angular speed at radius r = 0.0268m is

$\omega = \frac{v}{r} = \frac{0.031}{0.0268} = 0.114 rad/s$