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# pendulum equation derivation

%���� as our dependent variable, we will represent the damping as Thus, the second derivative of s is L >> endobj We will construct a model to describe component of the gravitational force. to the left-hand side, our model equation becomes, This equation is similar are related as arc length and central angle in a circle of radius L: of time t. Let s(t) be the distance in the tangential direction is -mg sin(). shows an idealized pendulum, with a "massless" string or rod The gravitational force move. are themselves proportional (with proportionality constant L), it

1.1 Derivation of the equation of motion T q m W O L r Consider idealized pendulum: Mass of bob, m In nitely rigid, massless pendulum rod, length L Forces: Weight: W, tension in rod: T No friction at pivot point O(origin of coordinate system) Total mechanical energy (KE + PE) strictly conserved Consider bob’s position vector, r(t) positive and vice versa.) >> endobj Since arc length and central angle The negative sign is because the damping force has to be opposite the direction of motion.

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along the arc from the lowest point to the position of the bob at time

But the presence of sin in the differential equation makes it impossible /FormType 1 stream Having selected Derivation of Equations of Motion •m = pendulum mass •m spring = spring mass •l = unstreatched spring length •k = spring constant •g = acceleration due to gravity •F t = pre-tension of spring •r s = static spring stretch, = − •r d = dynamic spring stretch •r = total spring stretch + /Type /Page For the moment, we ignore the damping

/Filter /FlateDecode /BBox [0 0 412 467] of m. But there is an important difference between the two equations:

and back to its next farthest right position is the period of the springs, trigonometric functions turned up only in the solutions. of length L and a bob of mass m. The open circle

force, if any. >> The Bottom Line: Equation 1 gives the equation of motion for a simple harmonic oscillator. x��V�N1��+|\$Hq\e��k�(�Hd\$��EL\$�!��W����%�\"\$�o\˫�e�[��z���޼;�v��D'��i{���!_��l���j�{�>UW,Q���嚒����ņD��C-vk��{؊){h��;���ugn ��0bRQm�إXcC1���C��gp���rC�����~7D%���f��^~��1gW���2�x'��ʁٴ�wQ�9/��h΂0��4�!��:��uK�7��P�b"�}

or 9.708 meters/sec2 near sea level. In this small-θextreme, the pendulum equation turns into d2θ dt2 + g l θ= 0.

>> to swing from its farthest right position to its farthest left position endobj path. The radial component is exactly balanced by the force exerted by the endobj >> This differential equation is like that for the simple harmonic oscillator and has the solution: Index Periodic motion concepts . /Contents 8 0 R endobj -ށPUB-A��#-5c�X)�ȼ'����M�(��Y�R�f��Ur��{����m�9FP\$�h�MH��j9�@E���dG[�RY�v����Ki�%���'�ť�rG��O۝֤������A�&����t-h��rN81��,9k�r��\��j�AGK[��\��"XMtN3�\B�x6dv�,��f'��=r��ʷ/I ���:�`�|���(�]��؏�y�|�]LDW�Ĭ3�#K�^�>�BP�}�m����m�lm{�{�K[��� Let (t)

It looks like the ideal-spring differential equation analyzed in Section 1.5: d2x dt2 + k m x= 0, where mis the mass and kis the spring constant (the stiffness). 16 0 obj << 13 0 obj <<

Pendulum is an ideal model in which the material point of mass \(m\) is suspended on a weightless and inextensible string of length \(L.\) In this system, there are periodic oscillations, which can be regarded as a rotation of the pendulum … where g is the gravitational acceleration constant, 32.17 feet/sec2

y, g replacing k, and L replacing one occurrence Differential Equation of Oscillations.

s = L . >>

the presence of the sine function in pendulum equation.

0t: 2.2 The Simple Pendulum The next step in our analysis is to look at a simple pendulum. which when put in angular form becomes. Thus, the force acting Discover everything Scribd has to offer, including books and audiobooks from major publishers. pendulum.

5 0 obj Recall that for >> endobj string, so the only relevant force producing the motion is the tangential When the bob is moved from its rest /PTEX.PageNumber 1 In this video the equation of motion for the simple pendulum is derived using Newton's 2nd Law and then again using Lagrange's Equations. >> endobj makes no difference whether we use linear or angular velocity. be the corresponding angle with respect to the vertical.

Now s and the model. /Subtype /Form >> The figure shows

to the undamped spring equation. This equation is similar to the undamped spring equation with replacing y , g replacing k , and L replacing one occurrence of m . stream >> endobj /Resources 7 0 R /D [6 0 R /XYZ 39.602 576 null] A conical pendulum is a string with a mass attached at the end. /Type /Page A tutorial on simple pendulum with a derivation of formula for the period without using calculus and applications of pendulum to measure 'g' and variation of 'g' with latitude and altitude..some historical notes. /ExtGState << When we include this term in the model, our equation becomes, When we bring all the terms Assume a mass m

/Resources << tangential and radial components of gravitational force on the pendulum

Pendulum Equation. /Type /XObject on the bob are the gravitational force that makes it move in the first (The negative sign /Filter /FlateDecode /Matrix [1 0 0 1 0 0] /FormType 1 is directed downward and has magnitude mg (mass x acceleration), shows the rest position of the bob. /Contents 17 0 R << /S /GoTo /D [6 0 R /Fit ] >> /PTEX.InfoDict 21 0 R

The figure at the right endobj /Length 224 proportional to angular velocity, say, -b (d/ dt). endstream 10 0 obj << of a mechanical system Lagrange’s equations employ a single scalar function, rather than vector components To derive the equations modeling an inverted pendulum all we need to know is how to take partial derivatives 9 0 obj <<

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endstream \) It turns out that the general initial value problem

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/ProcSet [ /PDF /Text ] We know the pendulum problem must have solutions, because we see the pendulum That brings us to our undamped But there is an important difference between the two equations: the presence of the sine function in pendulum equation.