Solve Pendulum Eq'n: λ≠0, μ=0 | Angular Freq ωd & Eigenfreq ω0

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In summary, the conversation discusses a homework problem involving a swinging pendulum and its equation. The goal is to solve the equation in two different cases and use the solutions to derive a relation between eigenfrequency, damping constant, and angular frequency. The individual mentions being confused about the angular frequency term in the equation and requests help, but then updates to say the problem has been solved.
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abotiz
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My assignment is in Swedish so I will try to Translate as good as I can ( if someone knows swedish you can visit the link http://physics.gu.se/~sjogren/FYP100/PM_svaengning.pdf )

Homework Statement



We have a swinging Pendulum which equation is given by :

Equation.jpg
attachment.php?attachmentid=18428&stc=1&d=1239580563.jpg


The variables κ, λ and μ are positive constants which are characteristic of the swinging system. θ is the Pendulum angular deviation from the equilibrium position. I = inertia moment

The Question :

Solve the equation in 2 cases

Case 1 : λ=μ=0

Case 2 : λ≠0, μ=0

The Purpose :

Use the Solutions to derive a realtion between the eigenfrequency ω0, damping constant [tex]\gamma[/tex] and angular frequency ωd for the damping Oscillation.

2. The attempt at a solution

Many, and i do not know what to do, because the angular frequency ω = dθ/dt which make the term μ*cos(ω*t) the whole equation into a separable equation ? Iam confused

I need help with this one. A solution to this would be appreciated!

Thanks
 
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  • #2
Never mind.
Solved!
 

FAQ: Solve Pendulum Eq'n: λ≠0, μ=0 | Angular Freq ωd & Eigenfreq ω0

What is the difference between angular frequency and eigenfrequency in the context of solving the pendulum equation?

Angular frequency (ωd) refers to the frequency at which the pendulum oscillates due to an external force. Eigenfrequency (ω0), on the other hand, is the natural frequency of the pendulum without any external forces acting on it. In other words, ω0 represents the frequency at which the pendulum would oscillate if left to its own devices, while ωd takes into account any external forces that may affect the oscillation.

How do the values of λ and μ affect the solution of the pendulum equation?

λ and μ are coefficients in the pendulum equation that can affect the stability and behavior of the pendulum's motion. λ represents the damping coefficient, which determines the rate at which the pendulum's energy dissipates due to friction or other external factors. μ is the coefficient of the driving force, which can affect the amplitude and frequency of the pendulum's oscillation. In general, larger values of λ and μ will result in more damped and less predictable oscillations.

Can the pendulum equation be solved analytically or does it require numerical methods?

In most cases, the pendulum equation cannot be solved analytically and requires numerical methods to find solutions. This is because the equation is a second-order differential equation, which can be difficult to solve using traditional algebraic methods. However, for certain simplified scenarios, such as a simple harmonic oscillator, an analytical solution may be possible.

How does the value of ωd compare to ω0 in terms of the pendulum's motion?

In general, ωd will be larger than ω0 since it takes into account the additional effects of external forces. This means that the pendulum will oscillate at a faster rate and with a larger amplitude for ωd compared to ω0. However, if the damping coefficient (λ) is high, the difference between ωd and ω0 may be small as the external forces are being counteracted by the dissipating energy.

What are some real-life applications of solving the pendulum equation?

The pendulum equation can be used to model and understand the behavior of various oscillating systems, such as a swinging pendulum or a vibrating guitar string. It is also used in fields such as engineering, physics, and astronomy to study the motion of objects and systems. In addition, the pendulum equation has been used to develop various technologies, such as clocks and seismometers, that rely on precise and predictable oscillations.

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