Normal modes of Mass and two Spings

This will result in an additional term in the EOM. In summary, the problem involves a mass confined to a flat surface with two identical springs attached to it. The goal is to find the normal modes of the system when the length of the springs is less than the distance between their fixed ends. The EOM takes into account the constant tension provided by the stretched springs and the additional tension when the mass is displaced from equilibrium. However, motion in the vertical direction must also be considered.
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Homework Statement



Consider a mass M whose motion is confined to a flat, smooth two-dimensional surface. Label the locations in 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 of its ends fixed to the point (-L, 0) and the other spring has one of its ends fixed to the point (L, 0). Find the normal modes of this system when ℓ < L.

Homework Equations



F=ma , T =-kx

The Attempt at a Solution



Well first I've my goal is to find the EOM of the system then plug in the solution to the EOM to find the normal modes. Since ℓ < L , then the springs are stretched in equilibrium and provide constant tension of T = -2k(L- ℓ). Taking into account when the mass is displaced a distance x from equilibrium also provides additional tension T = -2kx. Therefor my EOM is

ma = -2k(L- ℓ) + (-2kx)

However when I plug in a solution of the form x(t) = Acos(wt) it doesn't seem easy to solve.
Can someone please tell me if I am on the right track and if my EOM is correct?
 
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  • #2
You also need to consider motion in the vertical direction.
 

FAQ: Normal modes of Mass and two Spings

What are normal modes of mass and two springs?

Normal modes of mass and two springs refer to the natural oscillations or vibrations that occur when a mass is attached to two springs and released from a displaced position. These modes are characterized by specific frequencies and corresponding shapes of the oscillations.

How do mass and stiffness affect the normal modes?

The mass and stiffness of the system play a crucial role in determining the normal modes. A higher mass or stiffness will result in slower oscillations with lower frequencies, while a lower mass or stiffness will result in faster oscillations with higher frequencies.

What is the relationship between the natural frequency and the spring constants?

The natural frequency of a mass-spring system is directly proportional to the square root of the spring constant. This means that an increase in the spring constant will result in a higher natural frequency, while a decrease in the spring constant will result in a lower natural frequency.

Can normal modes be observed in real-life systems?

Yes, normal modes can be observed in many real-life systems, such as musical instruments, bridges, and buildings. These systems have natural frequencies that depend on their mass and stiffness, and when disturbed, they will vibrate at these frequencies.

What is the significance of normal modes in science and engineering?

Normal modes are important in understanding and analyzing the behavior of mechanical systems. They can help engineers design structures that can withstand certain frequencies of vibrations, and they are also essential in fields such as acoustics and seismology for predicting and mitigating the effects of vibrations.

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