Normal Modes - One Dimensional Oscillating Systems

[moolimanj]

Hi

Can someone help me with the following questions please (see attachment)?

I really need some help on the following:

i). Drawing a force diagram for each particle (I really hate drawing these).

As a guess for m1, am I right in thinking that H and N point up and W points down? But do I also need to add m3 to W (or would it be half m3?)

And for m3, would it be H1 and H2 up together with N, and W pointing downwards? Or would I also need to include m1 and m2?

ii). Can someone help me indicate the changes in forces exerted by the springs when the particles are displaced from their equilibrium positions? (the forces need to be expressed as vectors in terms of stiffness and displacements.)

iii) How do I derive the equation of motion of each particle?

 

[anathema]

Hello,

I would be happy to help you with your questions. First, let's start with drawing the force diagrams for each particle. The force diagram for m1 would include H pointing up, N pointing up, and W pointing down. You are correct in thinking that the weight of m3 would also be included in the downward force of W. It would be the full weight of m3, not just half.

For m3, the force diagram would include H1 and H2 pointing up, N pointing up, and W pointing down. You would not need to include m1 and m2 in the force diagram for m3.

Moving on to the changes in forces exerted by the springs when the particles are displaced from their equilibrium positions. The forces exerted by the springs are directly proportional to the displacement from the equilibrium position. This means that the farther the particle is displaced, the greater the force exerted by the spring will be. The forces can be expressed as vectors in terms of stiffness and displacement using the equation F = -kx, where F is the force, k is the stiffness constant, and x is the displacement from the equilibrium position.

To derive the equation of motion for each particle, you will need to use Newton's Second Law, which states that the sum of all forces acting on a particle is equal to the mass of the particle times its acceleration (ΣF = ma). You can use the force diagrams you drew to determine the forces acting on each particle and then plug them into the equation to solve for the acceleration. This will give you the equation of motion for each particle.

I hope this helps. Let me know if you have any further questions. Good luck!

 

[Moetasim]

Hello,

I would be happy to help you with these questions. First, let's start with the force diagram for each particle. For m1, you are correct in thinking that H and N point up, while W points down. However, you do not need to add m3 to W, as it is not directly connected to m1. For m3, H1 and H2 should also be pointing up, along with N, while W points down. You do not need to include m1 and m2 in this force diagram.

Moving on to the changes in forces exerted by the springs, these can be expressed as vectors in terms of stiffness and displacements. The force exerted by a spring is equal to its stiffness (k) multiplied by the displacement (x). So, for example, if m1 is displaced to the left, the spring connected to it will exert a force to the right, represented by a vector with magnitude kx. The same concept applies for the other particles and their corresponding springs.

To derive the equation of motion for each particle, you can use Newton's Second Law, which states that the sum of all forces acting on an object is equal to its mass (m) multiplied by its acceleration (a). So for m1, the equation of motion would be Fnet = m1a1 = -k1x1 (where x1 is the displacement from equilibrium). This can be rearranged to find the acceleration of m1, which can then be used to solve for its position as a function of time.

I hope this helps! Let me know if you have any further questions. Good luck with your work.