Significance of spring mass in SHM.

[hjalte]

Homework Statement

So, we are considering a spring-mass system, in which the mass at the end of the spring, M, is comparable to the mass of the spring, m.
Using Newtons laws, I have to calculate, how significant the mass of the spring is.

Homework Equations

Mass at the end of the spring, M.
Mass of the spring, m.
Spring constant, k.
Newtons second law, and Hooke's law.

The Attempt at a Solution

Actually, I have tried quite lot different approaches, but they don't seem to give me anything useful.
My latest attempt was to take a differential piece of mass of the spring, and calculate it's acceleration, in hope of getting something which i could integrate, but it didn't seem to work out.


I was asked this question by a high school student, whom I have to help writing a larger assignment.
I solved this problem rather easily using Lagrangian mechanics, but this is not available to the student, so I have to do it with Newtonian mechanics, which doesn't seem too easy.

I really appreciate any help I can get.

 

[LowlyPion]

Welcome to PF.

I would think if you know the length of the spring that you can calculate the kinetic energy along its length by integration on the basis of the velocities all along it's length. Then you can relate that to the kinetic energy of the attached mass at the end.

 

[thomate1]

Dear student,

The significance of the mass of the spring in a spring-mass system is crucial in understanding the dynamics of the system. This is because the mass of the spring affects the natural frequency of the system, which is the frequency at which the system oscillates in simple harmonic motion (SHM).

In SHM, the force acting on the mass is directly proportional to its displacement from the equilibrium position and is directed towards the equilibrium position. This is described by Hooke's law, which states that the force (F) is equal to the negative of the spring constant (k) multiplied by the displacement (x): F = -kx. The spring constant is a measure of the stiffness of the spring and is dependent on the material and geometry of the spring.

Now, when the mass of the spring is comparable to the mass at the end of the spring, the natural frequency of the system is affected. This is because the mass at the end of the spring not only has to overcome the restoring force of the spring, but it also has to move the mass of the spring itself. This results in a decrease in the natural frequency of the system compared to a system where the mass of the spring is negligible.

In order to calculate the significance of the mass of the spring, you can use Newton's second law, which states that the net force (F_net) acting on an object is equal to its mass (m) multiplied by its acceleration (a): F_net = ma. By considering a differential piece of mass of the spring and calculating its acceleration, you can integrate to find the total force acting on the entire spring. This will give you an idea of how much the mass of the spring affects the overall dynamics of the system.

I hope this helps in your assignment. Good luck!