A digital audio compact disc carries data along a continuous spiral track from the inner circumference of the disc to the outside edge. Each bit occupies 0.60 mm of the track. A CD player turns the disc to carry the track counter-clockwise above a lens at a constant speed of 1.22 m/s. Find the required angular speed (a) at the beginning of the recording, where the spiral has a radius of 2.10cm. (b) at the end of the recording, where the spiral has a radius of 6.10 cm. (c) A full- length recording lasts for 74 min, 33 s. Find the average angular acceleration of the disc as it spins from the outside at 2.10 cm to the inside at 6.10 cm.
t=74 min33 sec =4473 sá
ω1=v/R1 =1.22/0.021 = 58.1 rad/s
ω2=v/R2 =1.22/0.061 = 20 rad/s
ε= (ω2- ω1)/t =(20-58.1)/ 4473=-0.0085 rad/s²
To find the required angular speed at the beginning and end of the recording, we can use the formula for angular speed:
Angular speed (ω) = Linear speed (v) / radius (r)
(a) At the beginning of the recording, the radius (r) is 2.10 cm = 0.021 m, and the linear speed (v) is given as 1.22 m/s. Plugging these values into the formula, we get:
Angular speed at the beginning (ω) = 1.22 m/s / 0.021 m ≈ 58.1 rad/s
The required angular speed at the beginning of the recording is approximately 58.1 rad/s.
(b) At the end of the recording, the radius (r) is 6.10 cm = 0.061 m and the linear speed (v) is still 1.22 m/s. Plugging these values into the formula, we get:
Angular speed at the end (ω) = 1.22 m/s / 0.061 m ≈ 20.0 rad/s
The required angular speed at the end of the recording is approximately 20.0 rad/s.
(c) To find the average angular acceleration as the disc spins from the outside at 2.10 cm to the inside at 6.10 cm, we can use the formula:
Average angular acceleration (α) = (Final angular speed - Initial angular speed) / Time
First, let's find the final and initial angular speeds.
The final angular speed (ω_final) is the angular speed at the end of the recording, which we found to be approximately 20.0 rad/s.
The initial angular speed (ω_initial) is the angular speed at the beginning of the recording, which we found to be approximately 58.1 rad/s.
The time for the full-length recording is given as 74 min 33 s. Converting this to seconds, we get:
Time = 74 min × 60 s/min + 33 s = 4440 s + 33 s = 4473 s
Plugging these values into the formula, we get:
Average angular acceleration (α) = (20.0 rad/s - 58.1 rad/s) / 4473 s
= -38.1 rad/s / 4473 s
≈ -0.00852 rad/s²
The average angular acceleration of the disc as it spins from the outside at 2.10 cm to the inside at 6.10 cm is approximately -0.00852 rad/s².
To solve this problem, we need to utilize the relationship between linear speed, angular speed, and radius.
The linear speed (v) is given as 1.22 m/s, and we need to find the angular speed (ω) at two different radii: 2.10 cm and 6.10 cm.
(a) To find the angular speed at the beginning of the recording (r = 2.10 cm or 0.021 m), we can use the formula:
v = ωr
Rearranging the formula, we get:
ω = v / r
Plugging in the values, we have:
ω = 1.22 m/s / 0.021 m
ω = 58.1 rad/s
Therefore, the required angular speed at the beginning of the recording is 58.1 rad/s.
(b) To find the angular speed at the end of the recording (r = 6.10 cm or 0.061 m), we can again use the formula:
ω = v / r
Plugging in the values from above, we have:
ω = 1.22 m/s / 0.061 m
ω = 20.0 rad/s
Therefore, the required angular speed at the end of the recording is 20.0 rad/s.
(c) Finally, to find the average angular acceleration, we can use the formula:
α = Δω / Δt
Here, we are given the time for a full-length recording, which is 74 min 33 s. We need to convert this to seconds:
t = (74 min × 60 s/min) + 33 s
t = 4440 s
The change in angular speed (Δω) is the difference between the angular speeds at the two radii (ωend - ωstart):
Δω = 20.0 rad/s - 58.1 rad/s
Δω = -38.1 rad/s
Plugging the values into the formula, we have:
α = Δω / Δt
α = -38.1 rad/s / 4440 s
α ≈ -0.00859 rad/s²
Therefore, the average angular acceleration of the disc as it spins from the outside at 2.10 cm to the inside at 6.10 cm is approximately -0.00859 rad/s². The negative sign indicates that the disc is decelerating.