Problem 66 - University Physics with Modern Physics

Question

A computer disk drive is turned on starting from rest and has constant angular acceleration. If it took 0.0865 s for the drive to make its \(second\) complete revolution, (a) how long did it take to make the first complete revolution, and (b) what is its angular acceleration, in rad/s\(^2\)?

Step-by-Step Solution

Answer

(a) 0.0611 s; (b) 3386.52 rad/s².

Step 1: Understanding the Problem

We are given that a disk drive starts from rest with constant angular acceleration and takes 0.0865 s to complete its second revolution. We need to calculate the time taken for the first revolution and the angular acceleration.

Step 2: Using the Angular Kinematics Formula

The equation for angular displacement under constant angular acceleration is \( \theta = \omega_i t + \frac{1}{2} \alpha t^2 \). Since it starts from rest \( \omega_i = 0 \) and the angle for one revolution is \( 2\pi \) radians.

Step 3: Writing Equations for First and Second Revolutions

For the first revolution, \( 2\pi = \frac{1}{2} \alpha t_1^2 \). For the first and second revolutions combined, \( 4\pi = \frac{1}{2} \alpha t_2^2 \). We know \( t_2 = t_1 + 0.0865 \) s.

Step 4: Solving for Angular Acceleration

We can solve the equations simultaneously to find \( \alpha \), the angular acceleration. Substituting \( t_2 = t_1 + 0.0865 \) into the second equation: \( 4\pi = \frac{1}{2} \alpha (t_1 + 0.0865)^2 \).

Step 5: Simplifying and Solving for \( t_1 \)

From the first equation: \( \alpha = \frac{4\pi}{t_1^2} \). Substitute into the combined equation gives: \( 4\pi = 2\pi (\frac{t_2^2}{t_1^2}) \), solve for \( t_1 \) gives \( t_1 \approx 0.0611 \) s.

Step 6: Calculating Angular Acceleration

Using \( t_1 \approx 0.0611 \) s in the first formula for \( \alpha \), \( \alpha = \frac{4\pi}{(0.0611)^2} = 3386.52 \text{ rad/s}^2 \).

Key Concepts

Constant Angular Acceleration

Understanding constant angular acceleration is crucial to solving this problem. This term describes how an object's rotational speed changes at a constant rate over time. In layman's terms, it means that with each passing second, the disk drive spins faster and faster by the same amount.
For example, if the angular acceleration of the disk is 3386.52 rad/s two, as calculated in the solutions, it means that the rotational speed increases by this amount every second.

• Constant angular acceleration implies predictable rotational motion.
• It allows us to use simple equations to solve complex problems.

This concept is akin to linear acceleration in everyday experiences, like a car speeding up at a constant rate.

Angular Kinematics

Angular kinematics involves the equations used to describe rotational motion, similar to linear motion equations used for objects moving in a straight line. Here, however, we're dealing with angles, angular speed, and angular acceleration instead of distances, speeds, and linear accelerations.
The main formula used in this problem is:\[\theta = \omega_i t + \frac{1}{2} \alpha t^2\]where:

• \(\theta\): Angular displacement (in radians)
• \(\omega_i\): Initial angular velocity (\(0\) in this problem as it starts from rest)
• \(\alpha\): Angular acceleration
• \(t\): Time

This formula is essential to understanding how the drive completes its revolutions, beginning with zero speed and gradually speeding up.

Revolution Timing

Revolution timing refers to the time taken for the disk drive to complete one full spin or revolution. In this scenario, we’re examining the timing of both the first and second revolutions.
To find out how long it takes to reach one complete rotation, we need to consider its angular kinematics. The first complete revolution timing calculation uses \(2\pi = \frac{1}{2} \alpha t_1^2\) to find that it took about 0.0611 seconds.

• The second revolution takes 0.0865 seconds, as given.
• The difference between the two provides insights into how acceleration impacts timing.

By comparing these times, we can learn about the disk’s increasing speed during startup.

Angular Displacement

Angular displacement is the change in position of a point moving along a circular path. It's similar to how linear displacement measures change in position along a straight line, but in a circle, this change is expressed in radians.
For circular revolutions:

• One complete revolution equals \(2\pi\) radians.
• Two complete revolutions equal \(4\pi\) radians.

As seen in the exercise, these values are integral to determining the time required and calculating angular acceleration. Angular displacement, observed as \(2\pi\) or \(4\pi\) radians, helps quantify how much the disk has rotated over time. It's the foundation for exploring motion in circular paths.