# Experiment: Pendulum with Spring - Simulate Oscillation Trajectory

• qartveli
In summary, the conversation discussed the potential energy and kinetic energy equations for a 2d pendulum with a spring and the possibility of simulating this experiment on a computer. The equations are complex, but a small angle approximation could be used for a simulation. There is also the possibility of doing a stability analysis to determine the most stable trajectories.
qartveli
Hello everybody :) I need some help in an experiment simulation :) If you will make a pendulum with a spring instead of a string or rope (nonelastic), then bend it over on θ angle and then let it go - it will start to oscillate around with different manner than a simple pendulum :) with centripetal force spring will get longer and longer, after T/4 spring will start to get shorter and etc... This difference in length will change the period of oscillation of a pendulum and the trajectory too. So can anybody make simulate this experiment in the computer and post the picture of trajectory?

Interesting question. If we had just a 1d spring, the potential energy would be given by: $V = \frac{1}{2} k Q^2$ (Where Q is the displacement from equilibrium). And for a 2d pendulum, the potential energy is: $V = -mgrcos(\theta)$. Now if we define the displacement from equilibrium to be $r-l$ (in other words, the change in length of the pendulum, where $l$ is simply a constant), and if we add the two potentials together, we would get a total potential:
$$V = \frac{1}{2}k(r-l)^2 - mgrcos(\theta)$$
Now I'm going to talk about the 2d case, because the equations are easier. So the kinetic energy of the object is given by:
$$KE = \frac{1}{2}m(\dot{r}^2 + r \dot{\theta}^2 )$$
And now, we can use the Euler-Lagrange equations to find out the laws of the system:
$$-grsin(\theta) = \frac{d(r \dot{\theta})}{dt}$$
$$mr \dot{\theta} - k(r-l) + mgcos(\theta) = m \ddot{r}$$
And there is also the equation for the conservation of energy, which simply says that the kinetic energy plus the potential energy is conserved.

So, the equations are a bit complicated. We could also make the small angle approximation, which would make $sin(\theta) \rightarrow \theta$ and $cos(\theta) \rightarrow 1 - \frac{1}{2} \theta^2$ But it would still look quite complicated.

You could use these equations for a simulation on computer, and that would show the kind of trajectory to expect. And maybe there is a way to do stability analysis, which would show that certain trajectories are more stable than others, I'm not sure..

## 1. How does the length of the pendulum affect the oscillation trajectory?

The length of the pendulum directly affects the period of oscillation, with longer pendulums having longer periods. This means that the oscillation trajectory of a longer pendulum will take longer to complete one full swing compared to a shorter pendulum.

## 2. What is the relationship between the mass of the pendulum and its oscillation trajectory?

The mass of the pendulum does not have a significant effect on the oscillation trajectory. It only affects the period of oscillation if the mass is drastically different from the weight of the spring, causing changes in the stiffness of the spring and thus altering the period.

## 3. How does the stiffness of the spring affect the oscillation trajectory?

The stiffness of the spring, also known as its spring constant, directly affects the period of oscillation. A higher stiffness means a shorter period and a more rapid oscillation trajectory, while a lower stiffness results in a longer period and a slower oscillation trajectory.

## 4. What factors can affect the accuracy of the simulated oscillation trajectory?

The accuracy of the simulated oscillation trajectory can be affected by various factors, including the accuracy of the measurements and inputs used in the simulation, the precision of the equipment used, and external factors such as air resistance and friction.

## 5. How can this experiment be applied in real-life scenarios?

The principles learned from this experiment can be applied in various fields such as engineering, physics, and even music. For example, the concept of a pendulum with a spring can be used to design and test the stability and performance of structures, or to understand the oscillation of sound waves in musical instruments.

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