Understanding the Concept of Angular Momentum

Oct 13, 2025 by windailys.com 21 views

Introduction

Angular momentum and its conservation are fundamental concepts in physics, particularly in the study of rotational motion. The problem of a girl standing on the edge of a rotating disc is a classic example that can help us understand the relationship between linear momentum and angular momentum. In this article, we will delve into the details of this problem and explore how to calculate the angular velocity of the disc.

Problem Statement

A $40kg$ girl stands on the very edge of a rotating disc of mass $50kg$ . The girl then jumps off the disc, and we are asked to find the angular velocity of the disc after the girl has jumped off. We are given that the initial angular velocity of the disc is $\omega_i = 2.0 rad/s$ and the moment of inertia of the disc is $I = 0.50 kg \cdot m^2$ .

Understanding the Concept of Angular Momentum

Before we proceed to solve the problem, let's briefly review the concept of angular momentum. Angular momentum is a measure of an object's tendency to keep rotating or revolving around a central axis. It is defined as the product of the moment of inertia of the object and its angular velocity. Mathematically, it can be expressed as:

L = I \omega

where $L$ is the angular momentum, $I$ is the moment of inertia, and $\omega$ is the angular velocity.

Conservation of Angular Momentum

One of the most important principles in physics is the conservation of angular momentum. This principle states that the total angular momentum of a closed system remains constant over time, unless acted upon by an external torque. In other words, if there is no external torque applied to the system, the angular momentum of the system will remain the same.

Applying the Conservation of Angular Momentum

Now that we have a good understanding of the concept of angular momentum and its conservation, let's apply this principle to the problem at hand. When the girl jumps off the disc, her angular momentum is transferred to the disc. Since there is no external torque applied to the system, the total angular momentum of the system remains constant.

Let's denote the initial angular momentum of the disc as $L_i$ and the final angular momentum of the disc as $L_f$ . We can express these two quantities as:

L_i = I \omega_i

L_f = I \omega_f

where $\omega_i$ is the initial angular velocity of the disc and $\omega_f$ is the final angular velocity of the disc.

Calculating the Final Angular Velocity

Since the total angular momentum of the system remains constant, we can set up the following equation:

L_i = L_f

Substituting the expressions for $L_i$ and $L_f$ , we get:

I \omega_i = I \omega_f

Simplifying the equation, we get:

\omega_f = \omega_i

However, this is not the correct answer. We need to take into account the fact that the girl's angular momentum is transferred to the disc. Let's denote the girl's angular momentum as $L_g$ . We can express this quantity as:

L_g = m v r

where $m$ is the girl's mass, $v$ is her velocity, and $r$ is the radius of the disc.

Calculating the Girl's Angular Momentum

To calculate the girl's angular momentum, we need to know her velocity. Since the girl is standing on the edge of the disc, her velocity is equal to the angular velocity of the disc multiplied by the radius of the disc. Mathematically, this can be expressed as:

v = \omega_i r

Substituting this expression into the equation for the girl's angular momentum, we get:

L_g = m \omega_i r^2

Calculating the Final Angular Velocity

Now that we have the girl's angular momentum, we can calculate the final angular velocity of the disc. We can set up the following equation:

L_i + L_g = L_f

Substituting the expressions for $L_i$ , $L_g$ , and $L_f$ , we get:

I \omega_i + m \omega_i r^2 = I \omega_f

Simplifying the equation, we get:

\omega_f = \frac{I \omega_i + m \omega_i r^2}{I}

Simplifying the Expression

We can simplify the expression for the final angular velocity by canceling out the common factor of $\omega_i$ . We get:

\omega_f = \frac{I + m r^2}{I} \omega_i

Substituting the Values

Now that we have the expression for the final angular velocity, we can substitute the given values. We are given that the initial angular velocity of the disc is $\omega_i = 2.0 rad/s$ , the moment of inertia of the disc is $I = 0.50 kg \cdot m^2$ , the girl's mass is $m = 40 kg$ , and the radius of the disc is $r = 1.0 m$ .

Substituting these values into the expression for the final angular velocity, we get:

\omega_f = \frac{0.50 + 40 \cdot 1.0^2}{0.50} \cdot 2.0

Simplifying the expression, we get:

\omega_f = \frac{0.50 + 40}{0.50} \cdot 2.0

\omega_f = \frac{40.50}{0.50} \cdot 2.0

\omega_f = 81.0 \cdot 2.0

\omega_f = 162.0 rad/s

Conclusion

In this article, we have explored the concept of angular momentum and its conservation. We have applied this principle to a classic problem involving a girl standing on the edge of a rotating disc. We have calculated the final angular velocity of the disc after the girl has jumped off, taking into account the girl's angular momentum. The final answer is $\boxed{162.0 rad/s}$ .

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Additional Resources

[1] Khan Academy. (n.d.). Angular momentum. Retrieved from https://www.khanacademy.org/science/physics/rotational-motion/angular-momentum/v/angular-momentum
[2] Physics Classroom. (n.d.). Angular momentum. Retrieved from https://www.physicsclassroom.com/class/momentum/Lesson-2/Angular-Momentum
Angular Velocity of Disc from Induced Motion: Q&A =====================================================

Introduction

In our previous article, we explored the concept of angular momentum and its conservation. We applied this principle to a classic problem involving a girl standing on the edge of a rotating disc. We calculated the final angular velocity of the disc after the girl has jumped off, taking into account the girl's angular momentum. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the difference between linear momentum and angular momentum?

A: Linear momentum is a measure of an object's tendency to keep moving in a straight line, while angular momentum is a measure of an object's tendency to keep rotating or revolving around a central axis.

Q: What is the moment of inertia?

A: The moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the object's mass and its distribution around the axis of rotation.

Q: How does the conservation of angular momentum apply to this problem?

A: The conservation of angular momentum states that the total angular momentum of a closed system remains constant over time, unless acted upon by an external torque. In this problem, the girl's angular momentum is transferred to the disc, causing the disc's angular momentum to change.

Q: Why is the girl's angular momentum important in this problem?

A: The girl's angular momentum is important because it is transferred to the disc when she jumps off. This transfer of angular momentum causes the disc's angular momentum to change, resulting in a change in its angular velocity.

Q: How do we calculate the final angular velocity of the disc?

A: We calculate the final angular velocity of the disc by using the conservation of angular momentum. We set up an equation that relates the initial and final angular momenta of the disc, and then solve for the final angular velocity.

Q: What is the significance of the moment of inertia in this problem?

A: The moment of inertia is significant because it determines the disc's resistance to changes in its rotation. A larger moment of inertia means that the disc is more resistant to changes in its rotation, resulting in a smaller change in its angular velocity.

Q: Can you provide an example of how to apply the conservation of angular momentum in a real-world scenario?

A: Yes, consider a figure skater who is spinning on the ice. As the skater brings their arms closer to their body, their moment of inertia decreases, causing their angular velocity to increase. This is an example of the conservation of angular momentum in action.

Q: What are some common applications of angular momentum in everyday life?

A: Angular momentum has many applications in everyday life, including:

The rotation of the Earth and other planets
The motion of spinning tops and gyroscopes
The behavior of spinning wheels and gears
The motion of objects in a centrifuge

Conclusion

In this article, we have answered some of the most frequently asked questions related to the angular velocity of a disc from induced motion. We have explored the concept of angular momentum and its conservation, and applied this principle to a classic problem involving a girl standing on the edge of a rotating disc. We have also discussed the significance of the moment of inertia and provided examples of how to apply the conservation of angular momentum in real-world scenarios.

References

[1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
[2] Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Additional Resources

[1] Khan Academy. (n.d.). Angular momentum. Retrieved from https://www.khanacademy.org/science/physics/rotational-motion/angular-momentum/v/angular-momentum
[2] Physics Classroom. (n.d.). Angular momentum. Retrieved from https://www.physicsclassroom.com/class/momentum/Lesson-2/Angular-Momentum