On an open-reel tape deck, the tape is being pulled past the playback head at a constant linear speed of 0.372 m/s.
(a) If the radius of the film taken up so far (as shown in the upper part of the drawing) is 0.0576 m, find the angular speed of the take-up reel.
____ rad/s
(b) After 2.43 103 s, the take-up reel is almost full, as the lower part of the drawing indicates. The radius of film taken up so far is now 0.114 m. Find the average angular acceleration of the reel.
_____ rad/s2
Specify whether the acceleration indicates an increasing or decreasing angular velocity.
(a) Well, well, let's unwind this problem, shall we? We know that the linear speed of the tape is 0.372 m/s. Now, the linear speed of the tape is related to the angular speed of the take-up reel by the equation: v = ωr, where v is the linear speed, ω is the angular speed, and r is the radius.
So, by plugging in the given values, we have 0.372 m/s = ω * 0.0576 m. Solving for ω, we get:
ω = 0.372 m/s / 0.0576 m = ____ rad/s. Fill in the blank, circus master!
(b) Alright, continuing with our reel of fun, we are now given a new radius of 0.114 m and a time of 2.43 * 10^3 s. We need to find the average angular acceleration of the take-up reel. To do that, we can use the equation: ω f = ω i + αt, where ω f is the final angular speed, ω i is the initial angular speed (which is zero in this case), α is the angular acceleration, and t is the time.
Since the initial angular speed is zero, the equation simplifies to: ω f = αt.
We know that the final angular speed, ω f, is the angular speed we calculated in part (a). So, by plugging in the given values, we have: ____ rad/s = α * (2.43 * 10^3 s).
Solve for α, and don't forget to tell me whether the angular acceleration indicates an increasing or decreasing angular velocity. Don't want to leave me hanging, do you?
(a) To find the angular speed of the take-up reel, we can use the formula:
Angular speed (ω) = Linear speed (v) / Radius (r)
Given:
Linear speed = 0.372 m/s
Radius = 0.0576 m
Substituting the values into the formula, we have:
Angular speed (ω) = 0.372 m/s / 0.0576 m
Calculating this, we find:
Angular speed (ω) ≈ 6.458 rad/s
Therefore, the angular speed of the take-up reel is approximately 6.458 rad/s.
(b) To find the average angular acceleration of the reel, we can use the following formula:
Angular acceleration (α) = Change in angular speed (Δω) / Time taken (Δt)
Given:
Initial radius = 0.0576 m
Final radius = 0.114 m
Time taken = 2.43 × 10^3 s
First, we need to find the change in angular speed:
Change in angular speed (Δω) = Final angular speed - Initial angular speed
To find the initial angular speed, we can use the formula from part (a):
Initial angular speed = 0.372 m/s / (0.0576 m)
Calculating this, we find:
Initial angular speed ≈ 6.458 rad/s
Now, we can find the final angular speed using the final radius and linear speed:
Final angular speed = 0.372 m/s / (0.114 m)
Calculating this, we find:
Final angular speed ≈ 3.263 rad/s
Now, we can calculate the change in angular speed:
Change in angular speed (Δω) = 3.263 rad/s - 6.458 rad/s
Calculating this, we find:
Change in angular speed (Δω) ≈ -3.195 rad/s
Finally, we can find the average angular acceleration using the formula:
Angular acceleration (α) = (-3.195 rad/s) / (2.43 × 10^3 s)
Calculating this, we find:
Angular acceleration (α) ≈ -0.00131 rad/s^2
Therefore, the average angular acceleration of the reel is approximately -0.00131 rad/s^2. This negative value indicates a decreasing angular velocity.
To solve these problems, we need to use the fundamental relationships between linear and angular quantities.
(a) To find the angular speed of the take-up reel, we can use the relationship between linear and angular speeds. The linear speed of the tape (0.372 m/s) is equal to the product of the angular speed (ω) and the radius of the film taken up so far (r). Mathematically, this can be expressed as:
v = ω * r
Rearranging the equation, we can solve for ω:
ω = v / r
Substituting the values given, we have:
ω = 0.372 m/s / 0.0576 m
Calculating this expression, the angular speed of the take-up reel is approximately 6.458 rad/s.
(b) To find the average angular acceleration of the reel, we can use the relationship between change in angular velocity (Δω), time (t), and average angular acceleration (α). The formula for average angular acceleration is:
α = Δω / t
Since the reel starts at rest, the initial angular velocity (ω₀) is zero. Therefore, the change in angular velocity (Δω) is equal to the final angular velocity (ω) after a time period (t). We can use the relationship between linear and angular velocities to find ω.
ω = v / r
Substituting the values given, we have:
ω = 0.372 m/s / 0.114 m
Calculating this expression, the final angular velocity ω after 2.43 × 10³ seconds is approximately 3.263 rad/s.
Therefore, the average angular acceleration is:
α = (3.263 rad/s - 0 rad/s) / (2.43 × 10³ s)
Calculating this expression, the average angular acceleration of the reel is approximately 0.001342 rad/s².
The positive value of the average angular acceleration indicates an increasing angular velocity.