Question

The tape in a videotape cassette has a total length 219 m and can play for 2.4 h. As the tape starts to play, the full reel has an outer radius of 45 mm and an inner radius of 13 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

0.83 rad/s

Step-by-step explanation:

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

Let this radius be r, we know that 13 < r < 45 mm and r is somewhere so that the area of the donut-shaped object between 13mm and r is the same as the area of the donut-shaped object between r and 45mm

In math terms:

`A_r - A_i = A_o - A_r`

where
`A_r = \pi r^2` is the area at radius r, Ai and Ao are the inner and outer area of the empty and full reel, respectively

`\pi r^2 - 13^2\pi = 45^2 \pi - \pi r^2`

`r^2 - 169 = 2025 - r^2`

`2r^2 = 2025 - 169 = 1856`

`r^2 = 928`

or 0.03046 m

The rolling velocity of the cassette if it can play a total length of 219 m for 2.4 hours (2.4 * 60 * 60 = 8640 s)

v = 219 / 8640 = 0.02535 m/s

Therefore, the angular speed at both wheel is

`\omega = v/r = 0.02535 / 0.03046 = 0.83 rad/s`