Problem 22 - University Physics with Modern Physics

Question

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits \(10^{-7} \mathrm{m}\) deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 \(\mathrm{mm}\) and 58.0 \(\mathrm{mm}\) , respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 \(\mathrm{m} / \mathrm{s}\) . (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a \(\mathrm{CD}\) is 74.0 min. What would be the length of the track on such a maximum-duration \(\mathrm{CD}\) if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum-duration CD during its 74.0 -min playing time? Take the direction of rotation of the dise to be positive.

Step-by-Step Solution

Answer

Angular speeds are 50.0 rad/s (inner) and 21.55 rad/s (outer). Track length is 5550 m. Average angular acceleration is -0.00641 rad/s².

Step 1: Understand the Relationship Between Linear and Angular Speed

The linear speed \(v\) of a point on a rotating disc is related to its angular speed \(\omega\) by the equation \(v = r\omega\), where \(r\) is the radius at that point. The problem provides the linear speed and requires finding the angular speed at different radii.

Step 2: Find the Angular Speed at the Innermost Radius

To find the angular speed \(\omega\) at the innermost radius of 25.0 mm (which is 0.025 m), rearrange the formula: \( \omega = \frac{v}{r}\). Substitute \(1.25\, \text{m/s}\) for \(v\) and \(0.025\, \text{m}\) for \(r\):\[ \omega = \frac{1.25}{0.025} = 50.0\, \text{rad/s} \].

Step 3: Find the Angular Speed at the Outermost Radius

Using the same formula \( \omega = \frac{v}{r} \), substitute \(1.25\, \text{m/s}\) for \(v\) and the outer radius 58.0 mm (0.058 m) for \(r\):\[ \omega = \frac{1.25}{0.058} \approx 21.55\, \text{rad/s} \].

Step 4: Calculate the Total Length of the Track

The maximum playing time is 74 minutes, which is equivalent to \( 74 \times 60 = 4440\) seconds. The linear speed is constant at \(1.25\, \text{m/s}\), so the total track length \(L\) can be calculated as \( L = v \times t = 1.25 \times 4440 = 5550\, \text{m} \).

Step 5: Determine the Average Angular Acceleration

Average angular acceleration \(\alpha\) is determined by the change in angular speed over time. The initial angular speed is \(50.0\, \text{rad/s}\), and the final angular speed is \( 21.55\, \text{rad/s}\). The time period is 4440 seconds. \(\alpha = \frac{\omega_f - \omega_i}{t} = \frac{21.55 - 50.0}{4440} \approx -0.00641\, \text{rad/s}^2\). Since the acceleration is negative, this indicates deceleration.

Key Concepts

Angular Speed Calculation

When dealing with rotating objects, understanding how quickly they spin is essential. This speed is described as angular speed, denoted by \(\omega\). Angular speed tells us how much angular displacement an object covers per unit of time. For the CD in our problem, we are tasked with finding angular speeds at two different points on the disc: the innermost and outermost points of the spiral track.

Angular speed \(\omega\) can be computed using the relation with linear speed \(v\) and radius \(r\):

• \(\omega = \frac{v}{r}\)

At the innermost part (radius of 0.025 m), substituting the given linear speed of 1.25 m/s yields \(\omega = 50.0\, \text{rad/s}\).

Similarly, for the outermost radius (0.058 m), the same process gives \(\omega \approx 21.55\, \text{rad/s}\). Notice how the angular speed decreases as the radius increases. This behavior is typical for systems where linear speed is constant while the path radius varies.

Linear and Angular Relationship

The relationship between linear and angular quantities is a fundamental concept in rotations. Here, we explore the relationship involving linear speed \(v\), angular speed \(\omega\), and radius \(r\). This relationship is captured by the equation:

• \(v = r \omega\)

This equation reveals how linear speed is simply the product of the radius and the angular speed. It's like how far an edge of a disc travels in a straight line as the disc rotates.

For the CD's track, a constant linear speed of 1.25 m/s is maintained across varying radii. This consistency results in different angular speeds at different points. At smaller radii, fewer rotations per unit time (higher angular speed) are required to keep up the linear speed. In contrast, larger radii need fewer rotations (lower angular speed) to travel at the same linear velocity.

Recognizing this relationship helps in solving various rotational dynamics problems, providing insight into how objects behave when they spin.

Average Angular Acceleration

Angular acceleration, denoted by \(\alpha\), describes how an object's angular speed changes over time. It is an important factor in understanding how rotations evolve. The average angular acceleration can be calculated as:

• \(\alpha = \frac{\omega_f - \omega_i}{t}\)

Here, \(\omega_i\) is the initial angular speed, \(\omega_f\) is the final angular speed, and \(t\) is the time interval.

For the maximum-duration 74-minute CD, this calculation involves a change from an initial angular speed of 50.0 rad/s to a final angular speed of 21.55 rad/s. Over the 4440-second playtime, the average angular acceleration works out to approximately -0.00641 rad/s². The negative sign indicates a deceleration—meaning the CD spins slower as it plays out.

Understanding average angular acceleration is crucial for predicting and explaining motion in systems where rotational speed doesn't remain constant. It provides a quantitative measure of how quickly an object's rate of spin is changing.