(c) The Angular Velocity Of A Disc Of Mass 3 Kg And Radius 0.6 M Changes From 10 Rev/s To 15 Rev/s In this article, we explore the fascinating dynamics of rotational motion, focusing on how the angular velocity of a disc varies under different conditions. Understanding the change in angular velocity is crucial for engineers, physicists, and students studying rotational mechanics, as it provides insights into how objects spin, accelerate, and conserve energy during rotation.
In this comprehensive guide, we will analyze the problem involving a disc with a mass of 3 kg and a radius of 0.6 meters, which experiences a change in its angular velocity from 10 revolutions per second (rev/s) to 15 rev/s. We will delve into the concepts of angular velocity, angular acceleration, moment of inertia, and rotational kinetic energy, providing clear explanations and practical examples.
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Understanding Angular Velocity and Its Significance
What Is Angular Velocity?
Angular velocity (\(\omega\)) is a measure of how quickly an object rotates or spins around a fixed axis. It is defined as the rate of change of angular displacement with respect to time and is usually expressed in radians per second (rad/s).The relationship between revolutions per second (rev/s) and radians per second is given by:
\[
\omega = 2\pi \times (\text{rev/s})
\]
where:
- 1 revolution = \(2\pi\) radians
Importance of Angular Velocity in Rotational Mechanics
Angular velocity is essential in various applications:
- Designing rotating machinery like turbines and engines
- Understanding the dynamics of spinning objects
- Calculating rotational kinetic energy
- Analyzing angular acceleration and torque
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Initial and Final Conditions of the Disc
Given Data
The problem provides the following information:- Mass of the disc, \(m = 3\,kg\)
- Radius of the disc, \(r = 0.6\,m\)
- Initial angular velocity, \(\omega_i = 10\,rev/s\)
- Final angular velocity, \(\omega_f = 15\,rev/s\)
Converting Rev/s to Rad/s
To perform calculations in SI units, convert the initial and final angular velocities:\[
\omega_i = 10\,rev/s \times 2\pi\,rad/rev = 20\pi\,rad/s \approx 62.83\,rad/s
\]
\[
\omega_f = 15\,rev/s \times 2\pi\,rad/rev = 30\pi\,rad/s \approx 94.25\,rad/s
\]
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Calculating the Change in Angular Velocity
Angular Acceleration (\(\alpha\))
The angular acceleration describes how quickly the disc's angular velocity changes over time:\[
\alpha = \frac{\omegaf - \omegai}{t}
\]
However, since the problem does not specify the time duration, we focus instead on energy and torque considerations, or we assume a certain time interval for more detailed analysis.
Determining the Work Done or Energy Change
The change in the disc's rotational kinetic energy can be calculated as:\[
\Delta KE = \frac{1}{2} I (\omegaf^2 - \omegai^2)
\]
where \(I\) is the moment of inertia of the disc.
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Moment of Inertia of the Disc
Formula for a Solid Disc
The moment of inertia for a solid disc about its central axis is given by:\[
I = \frac{1}{2} m r^2
\]
Substituting the given values:
\[
I = \frac{1}{2} \times 3\,kg \times (0.6\,m)^2 = \frac{1}{2} \times 3 \times 0.36 = 0.54\,kg\,m^2
\]
Significance of Moment of Inertia
The moment of inertia quantifies how much torque is needed for a given angular acceleration. A larger \(I\) means the object resists changes in its rotational speed more strongly.---
Calculating the Change in Rotational Kinetic Energy
Using the earlier energy formula:
\[
\Delta KE = \frac{1}{2} \times 0.54\,kg\,m^2 \times \left( (94.25)^2 - (62.83)^2 \right)
\]
Calculations:
\[
(94.25)^2 \approx 8884.56
\]
\[
(62.83)^2 \approx 3947.84
\]
\[
\Delta KE = 0.27 \times (8884.56 - 3947.84) = 0.27 \times 4936.72 \approx 1332.52\,J
\]
Interpretation: The disc gains approximately 1332.52 joules of rotational kinetic energy as its angular velocity increases from 10 rev/s to 15 rev/s.
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Torque Required for the Change in Angular Velocity
Relationship Between Torque and Angular Acceleration
The torque (\(\tau\)) necessary to produce a certain angular acceleration is given by:\[
\tau = I \alpha
\]
If the time taken for the change in angular velocity, \(t\), is known, one can compute \(\alpha\):
\[
\alpha = \frac{\omegaf - \omegai}{t}
\]
and then:
\[
\tau = I \times \frac{\omegaf - \omegai}{t}
\]
Note: Without the time interval, we can only analyze the torque in terms of the angular acceleration.
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Practical Applications and Real-World Examples
Rotational Mechanics in Engineering
Understanding the change in angular velocity is fundamental in designing:- Electric motors
- Wind turbines
- Automotive flywheels
- Gyroscopes and stabilization systems
Sports and Recreation
Rotational motion principles are used to optimize performance in activities like:- Figure skating spins
- Baseball pitches
- Bicycle wheel dynamics
Physics Education and Demonstrations
Experiments involving discs spinning at different speeds help students grasp concepts of inertia, energy conservation, and torque.---
Conclusion
The change in the angular velocity of a disc from 10 rev/s to 15 rev/s signifies a substantial increase in rotational kinetic energy, requiring a specific amount of torque and energy input. The calculations demonstrate the importance of the moment of inertia and angular acceleration in analyzing rotational dynamics. Understanding these principles not only enhances theoretical knowledge but also provides vital insights for practical engineering applications, sports science, and physics education.---
Summary of Key Points
- Angular velocity measures how fast an object spins around its axis, expressed in rad/s or rev/s.
- Converting rev/s to rad/s involves multiplying by \(2\pi\).
- The moment of inertia for a solid disc is \(I = \frac{1}{2} m r^2\).
- The change in rotational kinetic energy depends on the initial and final angular velocities and the moment of inertia.
- Torque required for a change in angular velocity relates to angular acceleration and inertia.
By mastering these concepts, students and professionals can better analyze and design systems involving rotational motion, ensuring efficiency, safety, and innovation in various technological fields.
