Question

The tape in a videotape cassette has a total

length 249 m and can play for 1.8 h. As the
tape starts to play, the full reel has an outer
radius of 35 mm and an inner radius of 10 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

`w=19.76 \ rad/s`

Step-by-step explanation:

Circular Motion

Suppose there is an object describing a circle of radius r around a fixed point. If the relation between the angle of rotation by the time taken is constant, then the angular speed is also constant. If that relation increases or decreases at a constant rate, the angular speed is given by:

`w=w_o+\alpha \ t`

Where
`\alpha` is the angular acceleration and t is the time. If the object was instantly released from the circular path, it would have a tangent speed of:

`\displaystyle v_t=w.r`

We have two reels: one loaded with the tape to play and the other one empty and starting to fill with tape. They both rotate at different angular speeds, one is increasing and the other is decreasing as the tape goes from one to the other. We'll assume the tangent speed is constant for both (so the tape can play correctly). Let's call
`w_1` the angular speed of the loaded reel and
`w_2` that from the empty reel. We have

`w_1=w_(o1)+\alpha_1 \ t`

`w_2=w_(o2)+\alpha_2 \ t`

If
`r_f=35\ mm=0.035\ m` is the radius of the reel when it's full of tape, the angular speed for the loaded reel is computed by

`\displaystyle w_(01)=(v_t)/(r_f)`

The tangent speed is computed by knowing the length of the tape and the time needed to fully play it.

`t=1.8\ h=1.8*3600=6480\ sec`

`\displaystyle v_t=(x)/(t)=(249\ m)/(6480\ sec)=0.0384 \ m/s`

`\displaystyle w_(01)=(0.0384)/(0.035)=1,098 \ rad/s`

If
`r_e=10\ mm=0.001\ m` is the radius of the reel when it's empty, the angular speed for the empty reel is computed by

`\displaystyle w_(02)=(0.0384)/(0.001)=38.426 \ rad/s`

The full reel goes from
`w_(01)` to
`w_(02)` in 6480 seconds, so we can compute the angular acceleration:

`\displaystyle \alpha_1=(w_(02)-w_(01))/(6480)`

`\displaystyle \alpha_1=(38.426-1.098)/(6480)=0.00576 \ rad/sec^2`

The empty reel goes from
`w_(02)` to
`w_(01)` in 6480 seconds, so we can compute the angular acceleration:

`\displaystyle \alpha_2=(1.098-38.426)/(6480)=-0.00576 \ rad/sec^2`

So the equations for both reels are

`w_1=1.098+0.00576 \ t`

`w_2=38.426-0.00576 \ t`

They will be the same when

`1.098+0.00576 \ t=38.426-0.00576 \ t`

Solving for t

`\displaystyle t=(38.426-1.098)/(0.0115)`

`t=3240 \ sec`

The common angular speed is

`w_1=1.098+0.00576 \ 3240=19.76 \ rad/s`

`w_2=38.426-0.00576 \ 3240=19.76 \ rad/s`

They both result in the same, as expected

`\boxed{w=19.76 \ rad/s}`