Diego Alejandro L.

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Physics (Newtonian Mechanics)

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Question:

Which is the Equation that determines the Dynamics of a pendulum subjected to the gravitational force with the acceleration $$g$$ in the small amplitude approximation?

Diego Alejandro L.

Answer:

From the application of the second Newton's Law, we have that the sum of forces in the $$x$$-component and the $$y$$-component are: $(\sum F_{y}=T-mg\cos(\theta)=0$) $(\sum F_{x}=-mg\sin(\theta)=ma$) As we consider that the motion of the pendulum is circular, we can consider that $$a=l\ddot{\theta}$$, so the equation in the $$x$$-component becomes: $(-mg\sin(\theta)=ml\ddot{\theta}$) Rearranging this equation, we get: $(ml\ddot{\theta}+mg\sin(\theta)=0$) $(\ddot{\theta}+\frac{g}{l}\sin(\theta)=0$) The approximation of small amplitude implies that $$\sin(\theta)\approx \theta$$, then, the equation is: $(\ddot{\theta}+\frac{g}{l}\theta=0,$) which we can identify as a harmonic oscillator equation, independent of the mass of the pendulum, where the angular frequency is: $(\omega=\sqrt{\frac{g}{l}},$) from this, we can realize that the frequency of the pendulum just depends on its length, but not the mass, which is a manifestation of the First Newton's Law.

Calculus

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Question:

What are the first two terms of the expansion of the function $$\ln(1+x)$$ around $$x=0$$?

Diego Alejandro L.

Answer:

By the Taylor Theorem, we know that a function $$f(x)$$ can be expanded around the value $$x=x_{0}$$ following the next power series expansion: $$f(x)\approx f(x_{0})+\left.\frac{df}{dx}\right|_{x=x_{0}}(x-x_{0})+\frac{1}{2!}\left.\frac{d^{2}f}{dx^{2}}\right|_{x=x_{0}}(x-x_{0})^{2}+\cdots$$ $$f(x)\approx \sum_{k=0}^{\infty}\frac{f^{(k)}(x_{0})}{k!}(x-x_{0})^{k}$$ If we begin with the function $$f(x)=\ln(1+x)$$, we have: $$f'(x)=\frac{1}{1+x}$$, $$f''(x)=-\frac{1}{(1+x)^{2}}$$, $$f^{(3)}(x)=\frac{2}{(1+x)^{3}}$$ then we have: $$f(0)=0$$ $$f'(0)=1$$ $$f''(0)=-1$$ $$f^{(3)}(0)=2$$ Then, from the Taylor Formula, we get: $$\ln(1+x)\approx x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots$$ In general, we get: $$\ln(1+x)=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{x^{k}}{k}$$

Partial Differential Equations

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Question:

What kind of solutions are expected for the next equation $$\frac{\partial^{2} u}{\partial x^{2}}=\frac{1}{c^{2}}\frac{\partial^{2}u}{\partial t^{2}}$$ Subjected to homogeneous Dirichlet Conditions in one dimension, i. e. for a string in one dimension?

Diego Alejandro L.

Answer:

The solutions expected, given the Boundary Conditions are of Fourier type, that is to say, solutions of the form: $$u(x,t)=\sum_{m=0}^{\infty}\left\{A_{m}\sin\left(\frac{m\pi ct}{L}\right)+B_{m}\cos\left(\frac{m\pi ct}{L}\right)\right\}\sin\left(\frac{m\pi x}{L}\right),$$ where the constants $$A_{m}$$ and $$B_{m}$$ depend on initial conditions, that is to say, the initial form of the string and the velocity profile of the string.

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