To answer question 1, we need to break up the problem into parts as follows:
First: As the CD starts from rest and accelerates for 47 seconds, we can use the equation: θ = 1/2 * α * t^2 where
θ is the angle (in radians),
α is the angular acceleration
t is the time
Plugging in the values: θ = 1/2 * 3.8 * (47)^2 θ ≈ 4169 radians
Second: As the CD maintains a constant angular velocity for 170 radians, the angular velocity is constant. We can use the equation: θ = ω * t ,
where: ω is the angular velocity and t is the time.
Using the values we get: 170 = ω * t ω = 170/t
Third: As the CD decelerates to rest in 41 seconds, We can use the same equation as before, but with a negative acceleration: θ = 1/2 * (-α) * t^2 θ = 1/2 * (-3.8) * (41)^2 θ ≈ -3164 radians
Adding up the three parts, we get: 4169 + 170*ω - 3164 ≈ 2807 radians
Therefore, the CD spins a total of about 2807 radians.
- The total time is the sum of the time spent accelerating, at a constant velocity, and decelerating:
total = 47 + 170/ω + 41,
Plugging in the value we found for ω in part 1: total = 47 + 170/(170/t) + 41
total time ≈ 263 seconds
Hence, it takes about 263 seconds for the CD to complete the described motion.
2: The average angular velocity is simply the total angle (found in part 1) divided by the total time
ω_avg = θ / t_total ω_avg ≈ 10.7 radians per second
Therefore, the average angular velocity is about 10.7 radians per second.
The total number of rotations:
We can simply divide the total angle (found in part 1) by 2π:
rotations = θ / (2π) rotations ≈ 665 rotations
Hence, the CD spins about 665 rotations.
