91Ó°ÊÓ

The initial angular velocity and the angular acceleration of four rotating objects at the same instant in time are listed in the table that follows. For each of the objects (a), (b), (c), and (d), determine the fi nal angular speed after an elapsed time of 2.0 s. $$ \begin{array}{lcc} & \begin{array}{c} \text { Initial angular } \\ \text { velocity } \omega_{0} \end{array} & \begin{array}{c} \text { Angular } \\ \text { acceleration } \alpha \end{array} \\ \text { (a) } & +12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \text { (b) } & +12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \text { (c) } & -12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \text { (d) } & -12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \end{array} $$

Step by step solution

01

Understand the Angular Motion Equation

The final angular velocity \( \omega \) can be found using the equation: \( \omega = \omega_0 + \alpha t \), where \( \omega_0 \) is the initial angular velocity, \( \alpha \) is the angular acceleration, and \( t \) is the elapsed time. Here, \( t = 2.0 \) s.

02

Solve for Object (a)

For object (a), \( \omega_0 = +12 \) rad/s and \( \alpha = +3.0 \) rad/s². Substitute these values into the equation: \( \omega = 12 + 3.0 \times 2.0 = 12 + 6 = 18 \text{ rad/s} \).

03

Solve for Object (b)

For object (b), \( \omega_0 = +12 \) rad/s and \( \alpha = -3.0 \) rad/s². Substitute these values into the equation: \( \omega = 12 - 3.0 \times 2.0 = 12 - 6 = 6 \text{ rad/s} \).

04

Solve for Object (c)

For object (c), \( \omega_0 = -12 \) rad/s and \( \alpha = +3.0 \) rad/s². Substitute these values into the equation: \( \omega = -12 + 3.0 \times 2.0 = -12 + 6 = -6 \text{ rad/s} \).

05

Solve for Object (d)

For object (d), \( \omega_0 = -12 \) rad/s and \( \alpha = -3.0 \) rad/s². Substitute these values into the equation: \( \omega = -12 - 3.0 \times 2.0 = -12 - 6 = -18 \text{ rad/s} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Velocity

Angular velocity is like speed but for objects that are spinning. It's a way to say how fast something is rotating. Imagine a bike wheel spinning—angular velocity tells us how quickly it’s turning. Just like how velocity measures how fast you're moving in a straight line, angular velocity does the same but for angles around a point. It's calculated in radians per second (rad/s), which might sound fancy but is just a way of measuring rotation. In the context of our problem, we’re starting with an initial angular velocity, which is how fast the object is already spinning at the very beginning.

Angular Acceleration

Angular acceleration is all about how the speed of rotation, or angular velocity, is changing. Imagine you're on a merry-go-round, and it starts speeding up. That increase in speed is due to angular acceleration. It can either increase or decrease the speed of something spinning, depending on whether it’s positive or negative. In simple terms, if angular acceleration is positive, the object speeds up. If it's negative, the object slows down. In our exercise, each spinning object has a specific angular acceleration, measured in radians per second squared (rad/s²), which factors into determining how their angular velocities evolve over time.

Equation for Final Angular Velocity

This handy equation helps us find out what the angular velocity will be after some time has passed if we know the initial angular velocity and the angular acceleration. The formula is \( \omega = \omega_0 + \alpha t \). Here, \( \omega_0 \) is the initial angular velocity, \( \alpha \) is the angular acceleration, and \( t \) is the time duration we're looking at. You can think of \( \omega_0 \) as your starting speed, \( \alpha \) as the rate of speed change, and \( t \) as how long that change is happening. By using this equation, we can calculate the final angular velocity after 2 seconds for each object in our exercise. It’s like a recipe where you plug in the ingredients \( \omega_0 \), \( \alpha \), and \( t \) to get the outcome \( \omega \).

Rotational Kinematics

Rotational kinematics is like the circle-dance version of regular kinematics. It deals with the motion of rotating objects, using angular versions of speed, acceleration, and displacement. Just as linear kinematics involves studying the velocity and acceleration of objects moving in a straight line, rotational kinematics applies similar principles but to objects that are spinning or rotating. This branch of physics allows us to understand the motion and predict how rotating objects behave over time. It's essential for calculating the paths and speeds of any rotating body, from planets to wheels. In problems like ours, rotational kinematics is the framework that ties together angular velocity and acceleration to predict future motion.