the tape in a videotape cassette has a total
length 256 m and can play for 2.3 h. As the
tape starts to play, the full reel has an outer
radius of 38 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.
To find the common angular speed of both reels during the play, we need to understand a few concepts.
First, let's consider the tape in the videotape cassette. The tape is wound around two reels, with an outer radius of 38 mm and an inner radius of 13 mm. The difference between the outer and inner radii gives us the width of the tape, which is:
Width of the tape = Outer radius - Inner radius
Width of the tape = 38 mm - 13 mm
Width of the tape = 25 mm
Next, we need to convert the width of the tape from millimeters to meters:
Width of the tape = 25 mm = 0.025 m
Now, we know that the length of the tape is 256 m and it can play for 2.3 hours. We can use these values to calculate the linear speed of the tape:
Linear speed = Length of the tape / Time taken to play
Linear speed = 256 m / (2.3 h * 60 min/h * 60 s/min)
Linear speed = 256 m / 8280 s
Linear speed ≈ 0.03095 m/s (rounded to five decimal places)
The linear speed of the tape is the rate at which the tape moves past a fixed point. It is also equal to the product of the angular speed and the radius. In this case, both reels will have the same angular speed when the tape is rolling between them.
Linear speed = Angular speed * Radius
0.03095 m/s = Angular speed * 0.013 m (taking the inner radius)
From this equation, we can find the common angular speed:
Angular speed = Linear speed / Radius
Angular speed = 0.03095 m/s / 0.013 m
Angular speed ≈ 2.38 rad/s (rounded to two decimal places)
Therefore, the common angular speed of both reels during the play is approximately 2.38 rad/s.
the common radius will be (38+13)/2 = 25.5 mm
The tape speed is 256m/2.3h = 30.9 mm/s
Now you should be able to figure the angular speed, knowing the circumference and linear speed. Think of the units involved.