# Trouble with conservation of angular momentum in a system

• fluidistic
In summary, the problem involves a system consisting of a fixed sphere with a spring attached to its north pole and a particle of mass m at the end of the spring. The Lagrangian and equations of motion are derived using spherical coordinates and the problem also involves finding the conserved quantities and the effective potential in terms of theta. The final question involves determining the values of energy and angular momentum for a specific range of values of k.
fluidistic
Gold Member

## Homework Statement

The system consists of a -fixed on the ground sphere- in which we attach a spring of natural length 0 and constant k at its north pole. At the end of the spring there's a particle of mass m. Gravity acts vertically downward. We stretch the spring and give an impulse to the particle in the $\hat \phi$ direction. The mass moves without friction over the surface of the sphere.
1)Write down the Lagrangian of the particle and the equations of motions using spherical coordinates.
2)What are the conserved quantities? Gives an expression of them in function of the coordinates and their time derivatives. Find the effective potential in function of theta only.
3)If the motion of the particle is constrained for 30 °< $\theta$ < 45°, find the values of the energy and angular momentum. Determine the range of the values for k for this situation to happen.

## Homework Equations

L=T-V. A few more.

## The Attempt at a Solution

$v^2=\dot r^2+r^2 \sin \phi \dot \theta ^2+r^2 \dot \phi ^2$. But since r is constant, $\dot r=0$. Thus $T=\frac{m}{2}(r^2 \dot \theta ^2 \sin ^2 \phi +r^2 \dot \phi ^2 )$.
The potential energy is the gravitational one ($mgr\cos \theta$) plus the one of the spring ($\frac{k\theta ^2 r^2}{2}$).
This gives me the Lagrangian $L=\frac{m}{2}(r^2 \dot \theta ^2 \sin ^2 \phi +r^2 \dot \phi ^2 )- \left ( mgr \cos \theta + \frac{k}{2} \theta ^2 r^2 \right )$.
However I do not see any of the 3 generalized coordinates r, theta and phi as cyclic.
One would expect the linear momentum in the direction of $\hat r$ to be constant and worth 0 and $\dot \phi$ to be a constant (different from 0) so that the angular momentum in the $\hat \phi$ direction would be conserved. But for that to happen, I shouldn't have gotten a $\phi$ term in my Lagrangian. I should have gotten $\phi$ as a cyclic coordinate for its generalized momentum (corresponding here to the angular momentum) to be conserved. However I do not get this. What am I doing wrong?!

Last edited:
Also, I do not see how to find the effective potential in function of \theta only. For the 3rd question, I cannot start it until I manage to solve the first two.

## 1. What is conservation of angular momentum?

Conservation of angular momentum is a physical law that states that the total angular momentum of a closed system remains constant over time, regardless of any internal changes or external forces acting on the system.

## 2. Why is conservation of angular momentum important?

Conservation of angular momentum is important because it helps us understand and predict the behavior of rotating objects or systems. It also has many practical applications, such as in spacecraft navigation and engineering.

## 3. What is the role of torque in conservation of angular momentum?

Torque, or the measure of a force's ability to cause rotational motion, is a crucial component of conservation of angular momentum. In a closed system, the torque acting on one object must be offset by an equal and opposite torque acting on another object in order to maintain the system's overall angular momentum.

## 4. Can conservation of angular momentum be violated?

No, conservation of angular momentum is a fundamental law of physics and cannot be violated. However, in certain situations, it may appear to be violated due to external forces or frictional effects.

## 5. How does conservation of angular momentum apply to everyday life?

Conservation of angular momentum can be observed in many everyday activities, such as ice skating, spinning a top, or throwing a frisbee. It also plays a role in the motion of planets and galaxies in the universe.

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