To solve this problem, we need to determine the common angular speed of the two reels when the tape is playing.

Given data:

• Total length of the tape: $L = 192 \, \text{m}$
• Playing time: $T = 1.6 \, \text{h} = 1.6 \times 3600 \, \text{s} = 5760 \, \text{s}$
• Outer radius of the full reel: $R_{\text{outer}} = 44 \, \text{mm} = 0.044 \, \text{m}$
• Inner radius of the full reel: $R_{\text{inner}} = 11 \, \text{mm} = 0.011 \, \text{m}$

The task is to find the common angular speed when both reels have the same angular velocity.

Step 1: Calculate the total number of revolutions.

The total length of the tape is $192 \, \text{m}$, and it plays in $5760 \, \text{s}$. The tape moves from the inner reel to the outer reel, so the total number of revolutions made by the tape can be computed by dividing the total length by the circumference of the reel at the outer radius.

The circumference of the outer reel is given by: $C_{\text{outer}} = 2 \pi R_{\text{outer}} = 2 \pi \times 0.044 \, \text{m} \approx 0.276 \, \text{m}$

Now, the total number of revolutions is: $N = \frac{\text{Total length of the tape}}{\text{Circumference of outer reel}} = \frac{192 \, \text{m}}{0.276 \, \text{m}} \approx 696.7 \, \text{revolutions}$

Step 2: Calculate the total angular displacement.

To find the angular speed, we need the total angular displacement in radians. The total number of revolutions corresponds to an angular displacement of $2 \pi \times N$ radians. Therefore: $\theta_{\text{total}} = 2 \pi \times 696.7 \, \text{revolutions} = 4384.7 \, \text{radians}$

Step 3: Calculate the angular speed.

The angular speed is the rate of change of angular displacement with respect to time. Since the total angular displacement is $4384.7 \, \text{radians}$ and the tape plays for $5760 \, \text{s}$, the average angular speed is: $\omega_{\text{avg}} = \frac{\theta_{\text{total}}}{T} = \frac{4384.7 \, \text{radians}}{5760 \, \text{s}} \approx 0.761 \, \text{rad/s}$

Thus, the common angular speed of the two reels when they have the same angular speed is approximately: $\boxed{0.761 \, \text{rad/s}}$

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Would you like further clarification on any part of this solution? Here are some related questions:

1. How would the angular speed change if the total length of the tape increased?
2. What would happen to the angular speed if the playing time was shorter?
3. How can you calculate the angular velocity of the inner reel?
4. What factors influence the angular velocity of the reels in a videotape cassette?
5. How would the system behave if the tape was wound evenly across both reels from the start?

Tip: When dealing with rotating objects like reels, always remember to relate linear quantities (like length) to angular ones (like radians) through the relationship between linear and angular displacement.