91Ó°ÊÓ

The table that follows lists four pairs of initial and fi nal angular velocities for a rotating fan blade. The elapsed time for each of the four pairs of angular velocities is 4.0 s. For each of the four pairs, fi nd the average angular acceleration (magnitude and direction as given by the algebraic sign of your answer). $$ \begin{array}{lcc} & \text { Initial angular } & \text { Final angular } \\ & \text { velocity } \omega_{0} & \text { velocity } \omega \\ \text { (a) } & +2.0 \mathrm{rad} / \mathrm{s} & +5.0 \mathrm{rad} / \mathrm{s} \\ \hline \text { (b) } & +5.0 \mathrm{rad} / \mathrm{s} & +2.0 \mathrm{rad} / \mathrm{s} \\ \hline \text { (c) } & -7.0 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s} \\ \hline \text { (d) } & +4.0 \mathrm{rad} / \mathrm{s} & -4.0 \mathrm{rad} / \mathrm{s} \\ \hline \end{array} $$

Step by step solution

01

Understanding Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time. It can be calculated using the formula \( \alpha = \frac{\omega - \omega_0}{t} \), where \( \omega \) is the final angular velocity, \( \omega_0 \) is the initial angular velocity, and \( t \) is the time period in seconds.

02

Calculating for Case (a)

Given: \( \omega_0 = +2.0 \, \text{rad/s} \), \( \omega = +5.0 \, \text{rad/s} \), \( t = 4.0 \, \text{s} \). Substitute these values into the formula: \( \alpha = \frac{+5.0 - (+2.0)}{4.0} = \frac{+3.0}{4.0} = +0.75 \, \text{rad/s}^2 \).

03

Calculating for Case (b)

Given: \( \omega_0 = +5.0 \, \text{rad/s} \), \( \omega = +2.0 \, \text{rad/s} \), \( t = 4.0 \, \text{s} \). Use the formula: \( \alpha = \frac{+2.0 - (+5.0)}{4.0} = \frac{-3.0}{4.0} = -0.75 \, \text{rad/s}^2 \).

04

Calculating for Case (c)

Given: \( \omega_0 = -7.0 \, \text{rad/s} \), \( \omega = -3.0 \, \text{rad/s} \), \( t = 4.0 \, \text{s} \). Use the formula: \( \alpha = \frac{-3.0 - (-7.0)}{4.0} = \frac{+4.0}{4.0} = +1.0 \, \text{rad/s}^2 \).

05

Calculating for Case (d)

Given: \( \omega_0 = +4.0 \, \text{rad/s} \), \( \omega = -4.0 \, \text{rad/s} \), \( t = 4.0 \, \text{s} \). Use the formula: \( \alpha = \frac{-4.0 - (+4.0)}{4.0} = \frac{-8.0}{4.0} = -2.0 \, \text{rad/s}^2 \).

06

Conclusion

We have calculated the average angular accelerations for all cases: (a) +0.75 \( \text{rad/s}^2 \), (b) -0.75 \( \text{rad/s}^2 \), (c) +1.0 \( \text{rad/s}^2 \), (d) -2.0 \( \text{rad/s}^2 \). The signs indicate the direction of the acceleration: positive for speeding up and negative for slowing down.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Angular Velocity

Angular velocity is the measure of how quickly an object rotates around an axis. It's akin to linear velocity, but instead of traveling a distance on a straight line, an object travels through an angle. We express angular velocity, \( \omega \), in radians per second (rad/s). This unit is derived from considering the angle swept out by a rotating object per unit of time.
Understanding angular velocity is crucial because it allows us to assess how fast something is spinning. In the context of a rotating fan blade, each blade's angular velocity indicates how rapidly it completes a circle. For instance, if the angular velocity is 5 rad/s, it spans 5 radians every second.

• High angular velocity implies a fast spin.
• Positive angular velocity means the rotation is in the standard direction (counterclockwise).
• Negative angular velocity suggests rotation in the opposite direction (clockwise).

Knowing the initial and final angular velocities helps in calculating the change over a period, which is vital for determining angular acceleration.

Moment of Inertia

Moment of inertia is somewhat of a rotational equivalent to mass in linear motion. It represents an object’s resistance to changes in its rotational motion. Think of it as how hard it is to spin or stop spinning an object. The greater the moment of inertia, the harder it is to change the rotation of the object.
For a beginner's insight, consider a solid disc and a hoop with the same mass. The hoop's mass is farther from its axis compared to the disc's mass. This leads to a greater moment of inertia for the hoop even though both objects weigh the same. Thus, it is tougher to accelerate the hoop rotationally.

• Moment of inertia depends on the mass separation concerning the rotational axis.
• It is generally denoted by the symbol \( I \).
• Moment of inertia is crucial for calculations involving rotational motion. It often determines how an object will respond to applied torques.

In problems involving rotating objects like fan blades, the moment of inertia helps understand how quickly the blade can accelerate, given a torque.

Rotational Motion

Rotational motion is the movement of an object around a central point or axis. It's everywhere in our daily lives, from the spinning of a ceiling fan to the rotation of Earth. Understanding rotational motion involves addressing several parameters, such as angular velocity, angular acceleration, and moment of inertia.
Rotational motion can be thought of in parallel to linear motion. Just as in linear motion, where forces cause acceleration, in rotational motion, torques cause changes in angular velocity.

• Angular Displacement: The angle through which a point or line has been rotated in a specified sense about a specified axis.
• Angular Velocity: How fast the angle changes, much like speed in linear motion.
• Angular Acceleration: How quickly the angular velocity changes, analogous to acceleration in linear dynamics.

In the context of fan blades, rotational motion principles allow us to compute how long it takes for a fan to speed up or slow down, and how much energy is involved in these processes.