Two components of angular momentum conserved ? All three components are conserved?

Angular momentum is a fundamental concept in physics that describes the rotational motion of objects. When we talk about the conservation of angular momentum, we often refer to the components of angular momentum in a system. To clarify your question, let's break down the conservation of angular momentum and the conditions under which its components are conserved.

Understanding Angular Momentum

Angular momentum (\( \mathbf{L} \)) is defined as the product of an object's moment of inertia and its angular velocity. Mathematically, it can be expressed as:

• \( \mathbf{L} = \mathbf{r} \times \mathbf{p} \)
• Where \( \mathbf{r} \) is the position vector and \( \mathbf{p} \) is the linear momentum of the object.

Components of Angular Momentum

Angular momentum is a vector quantity, which means it has both magnitude and direction. In three-dimensional space, it can be broken down into three components:

• \( L_x \) - the component along the x-axis
• \( L_y \) - the component along the y-axis
• \( L_z \) - the component along the z-axis

Conservation of Angular Momentum

Angular momentum is conserved in a closed system where no external torques are acting. This means that if the net external torque (\( \tau \)) is zero, the total angular momentum of the system remains constant:

• \( \frac{d\mathbf{L}}{dt} = \mathbf{\tau} = 0 \Rightarrow \mathbf{L} = \text{constant} \)

Which Components Are Conserved?

In a closed system with no external torques, all three components of angular momentum (\( L_x \), \( L_y \), and \( L_z \)) are conserved. However, if there are external torques acting on the system, only the components of angular momentum that are aligned with the direction of the net torque will change. The other components may still be conserved.

Examples to Illustrate

Consider a spinning figure skater who pulls in their arms. As they do this, their moment of inertia decreases, and to conserve angular momentum, their rotational speed increases. In this case, the total angular momentum is conserved, and all three components are conserved as long as no external torques are acting on them.

On the other hand, if a spinning top is subjected to an external force, such as friction or a push, the angular momentum may change in the direction of that force. For instance, if a torque is applied about the z-axis, the \( L_z \) component may change, while \( L_x \) and \( L_y \) could remain constant if no torque acts in those directions.

Summary

In summary, the conservation of angular momentum depends on the presence or absence of external torques. In a closed system with no external influences, all three components of angular momentum are conserved. However, if external torques are present, only certain components may be conserved, depending on the direction of the applied torque. Understanding these principles is crucial for analyzing rotational dynamics in various physical systems.