Classical Mechanics, Coupled Harmonic motion

In summary, the problem involves finding the equations of motion for a system with two springs of relaxed lengths l1, l2 and two blocks with coordinates x1, x2. The solution can be separated into two problems: one for the motion of the center of mass and one for the "internal motion" described by x1 and x2. The normal modes of vibration can be found by assuming x, x1, and x2 are all in the form Ce^pt and solving for p. It is important to note that the motion of the center of mass should be taken into account before using this assumption.
  • #1
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Homework Statement



Set up the equations of motion for the system shown in Fig. 4.16. The relaxed lengths of
the two springs are l1, l2 . Separate the problem into two problems, one involving the motion
of the center of mass, and the other involving the "internal motion" described by the two
coordinates x1, x2. Find the normal modes of vibration.

Homework Equations


Figure 4.16 is attached

The Attempt at a Solution



The problem is already turned in and we have a test tomorrow but here is the general outline of my solution. So the center of mass of blocks 1 and 3 is

[tex]r_{13}= xm_3+(x-l_1-x_1)m_1[/tex]

and using Newton's law, and the fact that the force on this subsystem is -k_2x_2, we have

[tex](m_1+m_2)\ddot r_{13}= m_3\ddot x+(\ddot x-\ddot x_1)m_1=-k_2x_2[/tex]

and similarly with the system 2-3

the total force on just block 3 is easy so we have these three equations, assume x, x_1, and x_2 all have the form Ce^pt for DIFFERENT constants C. Substitute and solve for p.

Main question: are the equations I outlined above correct?
 

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  • #2
The variables are x,x_1,x_2 so I would define each of them to be positive going towards the right. You haven't done that, so it makes it a bit more confusing, but still technically correct.

You should try the form Ce^pt only after you have taken away the motion of the centre of mass. (I'm guessing you knew that, but they might give you marks for saying it explicitly, so you should probably say it).
 

1. What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion and behavior of physical bodies under the influence of forces. It is based on the laws of motion and gravitation described by Sir Isaac Newton.

2. What is coupled harmonic motion?

Coupled harmonic motion refers to the motion of two or more objects that are connected by a spring or elastic material, causing them to move in a coordinated way. This type of motion is characterized by a shared frequency and amplitude.

3. What are the equations of motion for coupled harmonic motion?

The equations of motion for coupled harmonic motion are based on Hooke's law, which states that the force exerted by a spring is directly proportional to the displacement of the object from its equilibrium position. The equations also take into account the mass and acceleration of each object.

4. What is the significance of resonance in coupled harmonic motion?

Resonance occurs when the frequency of an external force matches the natural frequency of a system. In coupled harmonic motion, resonance can cause the amplitude of the motion to increase significantly, potentially leading to destructive vibrations. It is important to understand and account for resonance in order to avoid damaging the system.

5. What are some real-life examples of coupled harmonic motion?

Coupled harmonic motion can be observed in many natural and man-made systems. Some examples include a child swinging on a swing set, a tuning fork vibrating when struck, and the motion of molecules in a solid material. It is also commonly used in engineering and design, such as in suspension systems for cars and in the construction of bridges and buildings.

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