147328 research outputs found
Sort by
Brachistochrone problem
No full textIn this Demonstration, parallel planes passing through the vertices of a regular polygon bound the media. The inradius of the polygon is 1. When you roll the polygon with the second slider, a particular vertex of the polygon moves along a path and passes through the boundary planes at the marked points. When a light ray follows these points, Snell's law is satisfied. You can see this by selecting "show angles".
Call the angle between the ray and boundary plane after refraction θ. The angle between the normal to the ray and vertical direction is 2θ. The speed of light in a medium is proportional to the square root of the height difference between the midpoint of the path in this medium and the initial position of the particle. This difference is equal to 1+cos(2θ) and √(1+cos(2θ)) = √2 cosθ . This proves that r is constant.
The first slider "sides" controls the number of sides of the polygon. As the number of sides goes to infinity, the path taken by light approaches the cycloid, which is the answer to the brachistochrone problemSine, cosine, Snell's law,conservation of mechanical energy, cycloidComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic
Convolution of two densities
No full textGraphs, Integrals, probability, density function (Gaussian, uniform, tent and parabolic)The convolution of two functions can be thought of as a measure of the overlap of the graphs as one graph is shifted horizontally across the other. Formally, if f and g are functions, the convolution of the two is the function (f*g)(t) = ∫f(t-τ)g(τ)dτ from -∞ to ∞
The plot shows f(t-τ), that is, f shifted by t units, in blue, g(τ) in purple, and the product of the two in gold. Thus the gray area is exactly the value of the convolution at t.
If X and Y are independent random variables with respective density functions f and g, then the density function of X+Y is the convolution of f and g. Interestingly, the convolution of two Gaussian densities is a Gaussian densityComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic
Optics: Law of Reflection
No full textBasics notions of reflection of the lightYou are standing up in front of a mirror placed two meters away. What size must the mirror be in order to let you see your whole body from top to bottom? How high is the top edge of the mirror? What happens if you are closer? And if you are taller or shorter?Componente Curricular::Ensino Médio::FísicaComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Físic
Melípona marginata -rainha
No full textVídeo que mostra as abelhas Melípona marginata em atividades dentro da colméia em torno da rainha. Essa espécie constrói potes de alimento em seus ninhos, os quais são utilizados como refúgios coletivos para rainhas virgensComponente Curricular::Ensino Fundamental::Séries Finais::Meio Ambient
Van Aubel's theorem for quadrilaterals
No full textKnowledge about Plane Geometry and PolygonsConstruct squares on the outside of the four sides of a quadrilateral and draw lines between the centers of the opposite squares. Then the two lines have the same length and are perpendicularComponente Curricular::Ensino Fundamental::Séries Finais::Matemátic
Multivariable epsilon-delta limit definitions
No full textDefinition of a limit of single-variable functions, two-variable functions, surfacesThe definition of a limit:
The expression lim x→a f(x) = L is an abbreviation for: the value of the single-variable function f(x) approaches L as x approaches the value a. More formally, this means that f(x) can be made arbitrarily close to L by making x sufficiently close to a, or in precise mathematical terms, for each real ε>0, there exists a δ>0 such that 0<│x-a│<δ→│f(x)-L│<ε. In other words, the inequalities state that for all x except a within δ of a, f(x) is within ε of L.
This definition extends to multivariable functions as distances are measured with the Euclidean metric.
In the figure, the horizontal planes 10±ε represent the bounds on f(x,y) and the cylinder is │x-a│=δ. No matter what ε is given, a δ is found (represented by the changing radius of the cylinder) so that all points on the surface z=f(x,y) inside the cylinder are between the two planesComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemátic
Vertical Pendulum Seismometer
No full textMechanics lawsA classical pendulum seismometer consists of a spring, a mass (black), and a damping device (light orange). These are all connected to a rigid frame that is fixed to the ground. When the ground moves, the mass is not able to move exactly in sync because of the inertia of the mass. The differential movement between mass and frame is recorded as a seismogram. The amplitude and phase differences between true ground motion and relative mass motion depend on the damping constant and the ratio of the frame-motion frequency to the eigenfrequency of the seismometer. The plot on the right shows these differences with the black dot indicating the current mass positionComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Físic
Twelve-Tone Equal-Tempered Scales
No full textWave's theoryIn this Demonstration, change the starting frequency to hear the 12-tone equal-tempered octave based on this toneComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Físic
The Vibrating String
No full textMechanics lawsUsing the locators, you can construct approximations to a polynomial of arbitrary degree or a piecewise continuous function on the interval 0 to π. The trigger starts the solution with no initial velocity and shows the evolution of the string as a function of time. This solution is built through the so-called d'Alembert solutions, which are a superposition of left and right traveling waves. The construction of these solutions can be explicitly demonstrated by only plotting right or left traveling waves (better seen on a larger interval). The evolving string is the superposition of both waves. The time evolutions of the first three Fourier modes of the solutions are shown on the left of the plotComponente Curricular::Educação Superior::Ciências Exatas e da Terra::Físic
Josephus problem
No full textThis Demonstration illustrates the classic Josephus problem. In a circle of n soldiers, every m^(th) soldier is executed. Which soldiers survive?Componente Curricular::Ensino Médio::Matemátic