Question

The tape in a videotape cassette has a total length 261 m and can play for 2.4 h. As the tape starts to play, the full reel has an outer radius of 48 mm and an inner radius of 11 mm. At some point during the play, both reels willhave the same angular speed.What is this common angular speed?Answer in units of rad/s.

Answer 1

Given that the total length of the tape, d = 261 m

The time taken is

$\begin{gathered} t=2.4\text{ h} \\ =2.4\times60\times60 \\ =8640\text{ s} \end{gathered}$

The linear speed will be

$v=\frac{d}{t}$

Substituting the values, the speed will be

$\begin{gathered} v=\frac{261}{8640} \\ =0.0302\text{ m/s} \end{gathered}$

Also, the inner radius is

$\begin{gathered} r_i=11\text{ mm} \\ =0.011m\text{ } \end{gathered}$

The outer radius is

$\begin{gathered} r_o=48\text{ mm} \\ =0.048\text{ m} \end{gathered}$

The radius will be

$\begin{gathered} r\text{ =}\frac{0.011+0.048}{2} \\ =0.0295\text{ m} \end{gathered}$

The angular speed will be given by the formula,

$\begin{gathered} \omega=\frac{v}{r} \\ =\frac{0.0302}{0.0295} \\ =1.023\text{ rad/s} \end{gathered}$

Thus, the common angular speed is 1.023 rad/s.