Problem 13 - College Physics

Question

DVDs. The angular speed of digital video discs (DVDs) varies with whether the inner or outer part of the disc is being read. (CDs function in the same way.) Over a 133 min playing time, the angular speed varies from 570 rpm to 1600 rpm. Assuming it to be constant, what is the angular acceleration (in rad/s' \(s\) ) of such a DVD?

Step-by-Step Solution

Answer

The angular acceleration is approximately 0.0135 rad/s².

Step 1: Understand the Problem

The problem provides us with the initial and final angular speed of a DVD and the total time in minutes. We need to calculate the angular acceleration, assuming it is constant. We will convert all units to rad/s before calculation.

Step 2: Convert Angular Speed to Radians per Second

First, convert the speeds from revolutions per minute (rpm) to radians per second (rad/s) using the conversion factor: \(1\,\text{rpm} = \frac{2\pi}{60}\,\text{rad/s}\). So the initial angular speed \(\omega_i = 570 \,\text{rpm} \times \frac{2\pi}{60}\) and the final angular speed \(\omega_f = 1600 \,\text{rpm} \times \frac{2\pi}{60}\).

Step 3: Calculate Angular Acceleration

Angular acceleration \(\alpha\) is defined as the change in angular velocity divided by the change in time. \[\alpha = \frac{\omega_f - \omega_i}{\Delta t}\]First, calculate \(\omega_i\) and \(\omega_f\) using the given values:\[\omega_i = 570 \times \frac{2\pi}{60} = 59.69\,\text{rad/s}\]\[\omega_f = 1600 \times \frac{2\pi}{60} = 167.55\,\text{rad/s}\]Next, convert the total time from minutes to seconds:\[\Delta t = 133\,\text{minutes} \times 60 = 7980\,\text{seconds}\]Finally, calculate \(\alpha\):\[\alpha = \frac{167.55 - 59.69}{7980} \,\text{rad/s}^2 = 0.0135 \,\text{rad/s}^2\]

Step 4: Review Your Result

Review the steps to ensure all conversions and calculations were performed correctly. The angular acceleration of the DVD is found to be approximately \(0.0135 \,\text{rad/s}^2\).

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates or revolves around an axis. It is an essential concept in physics, especially when dealing with rotating objects like DVDs or CDs. It's simpler than linear speed, which measures how fast something moves from one place to another. Instead, angular speed focuses on the frequency of rotation.
To determine the angular speed, you calculate how many rotations or revolutions occur in a given period. Thus, angular speed is expressed as revolutions per minute (rpm) or, more scientifically, radians per second (rad/s).
Using angular speed helps in understanding and predicting the behavior of rotating systems, such as the performance variation between the inner and outer parts of a DVD as the disc spins.

Radians Per Second

In physics, radians per second (rad/s) is a common unit of angular velocity. It measures how fast something spins. To visualize this, think of a circle. A radian is basically the angle created when the length of the arc is equal to the radius of the circle. Since a circle is 360 degrees or approximately 6.28 radians (2π radians), radians offer an efficient way to describe rotational movement with less confusion.
Using rad/s is beneficial because it fits seamlessly into calculations involving other physical quantities like time and distance. It's inherently connected with circular motion and thus suits real-world applications, for instance, where conversion from rpm to rad/s is often required. For instance, if a DVD player reads the disc at varying speeds, having the angular speed in rad/s simplifies the assessment of such dynamics.

Conversion Factors

Conversion factors play a crucial role in physics problems, especially when dealing with different units of measurement. When converting units, conversion factors help you easily translate from one unit, like rpm, to another, such as rad/s. This transformation is necessary to perform accurate calculations.
Consider the conversion from rpm to rad/s, which requires multiplying rpm by the factor \( \frac{2\pi}{60} \). This conversion factor accounts for both the conversion from revolutions to radians (multiplying by \(2\pi\)) and from minutes to seconds (dividing by 60).
By understanding and applying the right conversion factors, you can ensure your calculations are accurate regardless of the initial unit measurements. This skill is particularly useful in analyzing motion-related equations and solving physics problems efficiently.

Physics Problems

Physics problems often require breaking down complex concepts into more understandable terms. This approach is especially evident when calculating angular acceleration, which is a change in angular speed over time.
To solve these problems, start by clearly understanding what's being asked. Next, identify the known quantities and determine what needs to be calculated. It's crucial to convert all measurements into compatible units using conversion factors. Finally, with all information in hand, apply the right formulas step-by-step to reach the solution, like the formula for angular acceleration \( \alpha = \frac{\omega_f - \omega_i}{\Delta t} \).
Breaking down physics problems makes them more approachable and less intimidating, ultimately helping you develop a clear, logical path to the solution.