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
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
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.
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.