# What Are the Normal Modes of a Mass-Spring System in 2-D?

• macker1
In summary, the problem involves a mass attached to two identical springs on a smooth, flat, two-dimensional surface. The normal modes of the system can be found by solving uncoupled equations of motion for each coordinate. The position of the mass at any time can be determined using the normal modes and the given initial conditions.
macker1

## Homework Statement

A mass M is confined to move on a smooth, flat, two-dimensional surface. Label the locations on this surface using the Cartesian coordinates (x,y). The mass is attached to two identical springs, each of length l and spring constant k. One spring has one end fixed to the point (-L, 0) and the other spring has one end fixed to the point (L, 0)

1) Find the normal modes of this system when l<L.
2) At t=0, the mass is released at rest from the point (0.1L, -0.2L). What is the position of the mass for all subsequent times?

## The Attempt at a Solution

I used the Lagrangian method to find the equations of motion (see attachment). I'm sure there is something I could do to simplify these two equations by making some sort of approximation. But I'm not sure what to do. Any help would be appreciated. Thanks!

#### Attachments

• DEs.png
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[1]: https://i.stack.imgur.com/q6U9y.jpgFor 1), you can solve the equation of motion for the normal modes by solving two uncoupled equations of motion. The equations would take the form:m*x_ddot + k*(x-L) = 0m*y_ddot + k*(y+L) = 0where x and y are the coordinates of the mass, m is the mass of the object and k is the spring constant.By solving these equations, you will get the normal modes as a function of time:x(t) = A*cos(wt) + Ly(t) = B*sin(wt) - Lwhere A and B are constants determined from the initial conditions and w is the angular frequency of the system.For 2), you can use these normal modes to find the position of the mass at any subsequent time. Using the initial conditions given, you can determine the values of A and B. Then, you can plug in any time t to find the position of the mass. The position would be given by:x(t) = A*cos(wt) + Ly(t) = B*sin(wt) - L

## 1. What is the concept of "One Mass, Two Springs, 2-D Motion"?

The concept of "One Mass, Two Springs, 2-D Motion" refers to a physical system in which a single mass is attached to two springs and is allowed to move in two dimensions. This system is commonly used in physics experiments to study the dynamics of simple harmonic motion and the effects of varying spring constants and mass on the motion of the system.

## 2. How do the spring constants affect the motion of the system?

The spring constants determine the stiffness of the springs and, therefore, the force exerted on the mass. A higher spring constant will result in a stronger force and a smaller amplitude of motion, while a lower spring constant will result in a weaker force and a larger amplitude of motion.

## 3. What is the equation that describes the motion of the mass in this system?

The equation that describes the motion of the mass is given by x(t) = A cos(ωt + φ), where x is the displacement of the mass from its equilibrium position, A is the amplitude of motion, ω is the angular frequency, and φ is the phase angle. This equation is derived from the principles of simple harmonic motion.

## 4. How does the mass affect the motion of the system?

The mass affects the motion of the system by determining the inertia or resistance of the system to changes in motion. A larger mass will result in a slower motion and a smaller amplitude, while a smaller mass will result in a faster motion and a larger amplitude.

## 5. What can be learned from studying this system?

Studying this system allows scientists to understand the principles of simple harmonic motion and how varying factors, such as spring constants and mass, affect the motion of a system. It also has practical applications in fields such as engineering and robotics, where understanding the dynamics of motion is crucial.

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