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SHM Phasors This post is only an extension to the previous article on AC Phasors. The reader is advised to read the article on AC Phasors here  first, to gain an elementary idea on phasors. In this article, we consider phasors used in describing Simple Harmonic Motion. Simple Harmonic Motion: The motion governed by a force proportional to the displacement of the particle from the mean position and directed towards the mean position is called simple harmonic motion. i.e. for an ideal particle of mass `m`, we have, ` F = -kx` i.e.   ` m \frac{\text{d}^{2}x}{\text{dt}^{2}} = -kx ` Suppose we have the mass particle `m` attached to a spring that applies force according to the above equation. We stretch it from the mean position to some distance say `a` and then release it.  The differential equation of this motion is then, ` \frac{\text{d}^{2}x}{\text{dt}^{2}} = - \frac{k}{m}x` Since both `k` and `m` are positive quantities, we can let `\frac{k}{m} = w^{2}`, for some `w...

(Refresh the browser once more for the equations to load properly) Animating AC Phasors In this post, we try to get an intuitive feeling about special quantities called phasors, used to represent current and voltage in AC circuits. We will see the theory at work, learning about complex numbers, sinors, and how these are used to represent voltage and current. At the end, we'll be able to input our custom values in the animation! Sinusoids: Any quantity that varies in the form of a sine or a cosine function is called a sinusoid. When a sinusoidally time-varying voltage is applied across an electrical component we get a corresponding sinusoidally time-varying current which we call an Alternating Current (AC). The voltage applied is called the alternating voltage. In the general form if ` v(t) = V_{0} sin(ωt) `, then the current is in the form ` i(t) = I_{0} sin(ωt + φ) `, where ` φ ` is called the phase difference, created by the properties of the ele...

Newton-Leibniz Rule Newton-Leibniz's rule helps us to find the derivative of an integral, with respect to some variable, where the limits of integration are some functions of the same variable. While there are many variations of this rule (see note 1 below), we consider here one of the simplest forms. The Theorem: Let ` f(t) ` be a continuous integrable function and let `u(x)` and `v(x)` be some functions of x. Then the derivative of the integral of `f(t)` w.r.t. x is given by,  ` \frac{\text{d}}{\text{d}x}\int_{u(x)}^{v(x)} f(t) dt = f(v(x))\cdot v'(x) - f(u(x)) \cdot u'(x) ` Proof:  The proof for this is very simple. Let g(t) be the anti-derivative of f(t). Then we can write,  ` \int_{u(x)}^{v(x)} f(t) dt = g(t) |_{u(x)}^{v(x)} = g(v(x)) - g(u(x)) ` Now taking the derivative using the chain rule, we get,  ` \frac{\text{d}}{\text{d}x} [g(v(x)) - g(u(x))] = g'(v(x)) \cdot v'(x) - g'(u(x)) \cdot u'(x)   ` However, by our assumption since` g ` is the anti-de...

An Elegant Way to Expand The Inverse Tangent In this post we shall see a method to expand the inverse tangent function ` tan^{-1}(x) ` using an infinite geometric series. The infinite series equips us with a neat and easy to remember technique to find the expansion. A Geometric Series: A series in which a term is obtained by multiplying the previous term by a constant ratio is called a geometric series. This contant ratio is also called the common ratio `r`. For example consider these series: ` 2, 4, 8, 16, 32, ... ` `1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \frac{1}{625}, ... ` In the first series, `r = 2` and in the second series `r = \frac{1}{5}`. Sum of an Infinite Geometric Series: In general, any geometric series can be written as,  `a, ar^{1}, ar^{2}, ar^{3}, ar^{4}, ... ` An infinite series however can only be summed when the series converges i.e. only when `|r| < 1 `. When this is true the sum `S` is given by ` \frac{a}{1-r} ` i.e. ` S = a + ar^{1} + ar^{2} + ar^{3}...

Magnetic Moment of A Rotating Charged Rod Magnetic Moment: In general for any closed loop magnetic moment is given by formula `\vec{μ}= ni \vec{A}` , where ` \vec{A}` represents the area vector of the loop determined as usual by right hand thumb rule. Now, the primary problem for us is like this: Prob. A rod with charge density ` ρ = ρ_{0}\frac{x}{L} ` is rotated about its one end along an axis perpendicular to its length. Then find the magnetic moment of the setup. For a Revolving Charged Particle: Before going into the problem, it is important to realise that we have been given a charged rod, and not a simple wire loop wherein a current flows. However as the rod shall rotate, the charges will tranverse through space thus creating an apparent current. In any circular path, when a charged particle say q revolves it appears to create and average current,  `<i> = qf = \frac{q}{T} = \frac{qω}{2π}` Thus in that case,  ` μ = (\frac{qw}{2π})(π r^{2}) = \frac{qωr^{2}}{2}` F...