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 1.16 m/s. (d) Assuming the acceleration is constant, find the total angular displacement of the disc as it plays from the outside at 2.40 cm to the inside at 6.20 cm.

Question

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 1.16 m/s. (d) Assuming the acceleration is constant, find the total angular displacement of the disc as it plays from the outside at 2.40 cm to the inside at 6.20 cm.

Answer 1

To find the total angular displacement of the disc as it plays from the outside to the inside, we can use the formula for angular displacement:

θ = ωt + (1/2)αt^2

Where:
θ - Angular displacement
ω - Initial angular velocity
α - Angular acceleration
t - Time

First, let's find the initial and final positions of the track.

Initial position: r1 = 6.20 cm = 0.0620 m (inside edge)
Final position: r2 = 2.40 cm = 0.0240 m (outside edge)

Next, let's find the radius difference:

Δr = r1 - r2 = 0.0620 m - 0.0240 m = 0.0380 m

The bit length is given as 0.60 mm = 0.00060 m.

Now, we can find the total number of bits in the spiral track:

Number of bits = (track length) / (bit length)

The track length can be found using the circumference formula:

Track length = 2πr1 - 2πr2 = 2π (r1 - r2) = 2π * 0.0380 m

Finally, we can find the total number of bits:

Number of bits = (2π * 0.0380 m) / 0.00060 m

Now, we need to find the time it takes to play the track:

Time = (track length) / (linear speed)

Time = (2π * 0.0380 m) / 1.16 m/s

Once we have the time, we can find the angular acceleration:

α = (final angular velocity - initial angular velocity) / t

Since the angular velocity is constant, the acceleration is 0.

Now, we have all the necessary information to calculate the angular displacement:

θ = ωt + (1/2)αt^2

But since the acceleration is 0, the equation simplifies to:

θ = ωt

Substituting the values:

θ = (ω) * (2π * 0.0380 m) / 1.16 m/s

Note: We don't have the value for the angular velocity (ω) in the given question. To obtain the value for ω, we would need additional information, such as the time it takes for one revolution of the disc or the total playtime duration. Without that information, we cannot provide the exact value for the angular displacement.