How Do You Solve Equations of Motion for a Coupled Harmonic Oscillator?

[lylos]

Homework Statement

Two masses are connected via a spring. Write the equations of motion, solve them (x(t)), and find the normal mode frequencies.

Homework Equations

F=ma
F=-kx

The Attempt at a Solution

I set up two equations:
$$F_{1}=ma_{1} , F_{2}=ma_{2}$$
$$-k(x_{2}-x_{1})=m\ddot{x_{1}} , k(x_{2}-x_{1})=m\ddot{x_{2}}$$

Then I solved the 2nd force equation for $$x_{1}$$ and plugged it into the first force equation resulting in:
$$-\ddot{x_{2}}=\ddot{x_{1}}$$

I really don't know what to do now... Any help would be greatly appreciated!

 

[anathema]

Hello! You're on the right track with setting up the equations of motion. To solve them, you can use the method of separation of variables. This involves separating the variables (in this case, x and t) and then solving each part separately. So for the first equation, you would have:

-k(x_{2}-x_{1})=m\ddot{x_{1}}
-kx_{2}+kx_{1}=m\ddot{x_{1}}

Then you can separate the variables by dividing both sides by m and by x_{1}, giving you:

\frac{-k}{m}x_{2}+\frac{k}{m}x_{1}=\ddot{x_{1}}

Next, you can define a new variable, let's say y, such that y=\frac{k}{m}x_{1}. This allows you to rewrite the equation as:

\frac{-k}{m}x_{2}+y=\ddot{y}

You can do the same thing for the second equation, and then you will have two separate equations that you can solve for y and x_{2}. From there, you can substitute back in for x_{1} and x_{2} to get the solutions for x(t).

To find the normal mode frequencies, you can use the fact that the solutions will be sinusoidal functions with a certain frequency. So you can plug in the solutions for x(t) into the equations of motion and solve for the frequency, which will give you the normal mode frequencies. I hope this helps! Let me know if you have any other questions.

 

[Moetasim]

I would like to provide some guidance on how to approach this problem. First, it is important to understand the concept of a coupled harmonic oscillator. In this system, two masses are connected by a spring, which means that the motion of one mass affects the motion of the other and vice versa. This creates a situation where the equations of motion for each mass are coupled or interconnected.

To solve this problem, you can use the principle of superposition, which states that the total motion of the system is the sum of the individual motions of each mass. This means that you can solve for the motion of each mass separately and then combine them to find the overall motion.

To start, you can use the equations of motion for each mass, which you have correctly written as F=ma and F=-kx. However, it is important to note that these equations should be written in vector form, as the forces and displacements are in different directions for each mass. So, the equations should be written as F_1=m\ddot{\vec{x_1}} and F_2=m\ddot{\vec{x_2}}.

Next, you can use Newton's Second Law to write the forces in terms of the displacements and solve for the equations of motion. This will result in two second-order differential equations, one for each mass. You can then solve these equations using standard techniques for solving differential equations, such as separation of variables or using a characteristic equation.

Once you have solved for the equations of motion, you can find the normal mode frequencies by looking at the solutions for x_1 and x_2. These frequencies represent the natural frequencies at which the system will oscillate without any external forces acting on it.

In summary, to solve this problem, you will need to use the principle of superposition, Newton's Second Law, and standard techniques for solving differential equations. I hope this helps guide you towards finding a solution to this problem. Good luck!