91Ó°ÊÓ

In angular kinematics, angular acceleration represents how quickly an object's rotational velocity changes over time. Constant angular acceleration means that this change is steady and uniform, much like how gravity provides constant acceleration to a freely falling object. When a rotating system begins from rest, this concept helps in predicting how fast it will turn at any given point.

With constant angular acceleration, the rotation speed increases linearly with time. This property simplifies calculations because you don't need to worry about changes in the rate of increase. You can apply kinematic equations similar to those used in linear motion but adapted for rotational scenarios. These include equations for angular displacement, angular velocity, and time.

• Useful for predicting motion in circular or rotary systems like wheels, disks, and motors.
• Equations used: \(\theta = \omega_0 t + \frac{1}{2}\alpha t^2\), where \(\theta\) is the angular displacement, \(\omega_0\) is initial angular velocity, and \(\alpha\) is angular acceleration.

Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. It's analogous to linear displacement, measuring how far an object has traveled, but in the context of rotational motion.

In this exercise, the total angular displacement is given for each complete revolution. One complete revolution corresponds to an angular displacement of \(2\pi\) radians. Therefore, for the second revolution, the disk underwent a total of \(4\pi\) radians. This displacement can be determined using the kinematic equation, assuming constant angular acceleration.

• For the first complete revolution: \(\theta = 2\pi\) radians.
• For the second complete revolution: \(\theta = 4\pi\) radians.

To find the angular acceleration, we used details from the second complete revolution: the time taken (0.750 seconds) and the total angular displacement \(4\pi\) radians. Knowing that the system starts from rest simplifies the equation, as the initial angular velocity \(\omega_0\) is zero.

Substitute these known values into the equation:
\[4\pi = \frac{1}{2}\alpha (0.750^2)\]
Rearranging gives us:
\[\alpha = \frac{4\pi}{0.28125} \approx 44.63 \, \text{rad/s}^2\]
This calculation shows the uniform rate of increase in angular velocity from zero for this system.

• It's crucial to plug values correctly to avoid errors in finding \(\alpha\).
• Understanding the formula allows you to solve similar problems by just changing known variables.

Once the angular acceleration is calculated, you can determine how long different revolutions take. For this problem, once \(\alpha\) is known, finding the time for the first revolution is straightforward. We know from the problem that \(\theta = 2\pi\) radians.

Using the equation:
\[2\pi = \frac{1}{2}\alpha t_1^2\]
Substitute \(\alpha = 44.63\) into the equation:
\[t_1^2 = \frac{4\pi}{44.63}\approx 0.2815\]
Taking the square root provides the time for the first revolution:
\[t_1 \approx 0.53 \, \text{seconds}\]
These kinds of calculations help in understanding performance and timing in systems relying on rotational motion, such as disk drives or engines.

• Careful calculation ensures accurate results and understanding of rotational timelines.
• This principle can be reused across different rotational scenarios by adjusting figures.