Normal modes: Spring and pendulum

In summary, The conversation is about the calculation of kinetic energy in an exercise. The person speaking disagrees with the given solution and believes their own expression for ##T_2## is correct. However, they were not able to find any errors in their calculation. The other person agrees that the given expression is incorrect and suggests a simpler version for small oscillations.
  • #1
LCSphysicist
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Homework Statement
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Relevant Equations
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I was doing the exercise as follows:
oi.png

I am not sure if you agree with me, but i disagree with the solution given.
I was expecting that the kinect energy of the mass ##m## (##T_2##) should be $$T_2 = \frac{m((\dot q+lcos(\theta)\dot \theta)^2 + (lsin(\theta) \dot \theta)^2)}{2}$$
I could be wrong, of course, but i have tried to figure out my error and was not able to discover. So my guess is that the solution can be wrong.
 
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  • #2
Your expression for ##T_2## looks correct. For small oscillations it can be simplified a little.

The expression for ##T_2## in the solutions is not correct.
 
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1. What are normal modes in the context of spring and pendulum systems?

Normal modes refer to the different ways in which a spring or pendulum can oscillate or vibrate. These modes are characterized by specific frequencies and patterns of motion.

2. How are normal modes related to the natural frequency of a system?

The natural frequency of a system is determined by its physical properties, such as mass and stiffness. Normal modes correspond to the different natural frequencies of a system, with each mode having a specific frequency and associated pattern of motion.

3. How do normal modes affect the behavior of a spring or pendulum system?

The normal modes of a system determine how it responds to external forces or disturbances. If the system is excited at one of its natural frequencies, it will oscillate with a large amplitude. If the excitation frequency does not match any of the natural frequencies, the system will not exhibit large amplitude oscillations.

4. Can a system have more than one normal mode?

Yes, a system can have multiple normal modes, each with a different frequency and pattern of motion. The number of normal modes a system has is equal to the number of degrees of freedom it possesses.

5. How can normal modes be calculated for a spring or pendulum system?

Normal modes can be calculated using mathematical equations and principles of physics, such as the equations of motion and conservation of energy. These calculations can be complex, but there are also computer programs and simulations that can help determine the normal modes of a system.

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