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Finally, let us consider the problem of the pendulum, okay?

Â In which you'll be able to follow in the situation.

Â A mass M, suspending from the end of the rod of length L,

Â is swinging in a vertical plane.

Â And let theta of T be the angle,

Â measured in radian from the vertical with the right hand side to be positive.

Â Look at this picture.

Â Here you have a certain mass,

Â suspended from the rigid ceiling and make

Â a displacement in the right direction by the angle theta.

Â Then the length of the arc with

Â the vertical angle theta is equal to capital lambda is equal to L of theta.

Â There'll be now from the geometry,

Â this is the length of this displacement capital lambda,

Â which is equal to L times a theta,

Â because the theta is measure in radian.

Â On the other hand, the angular acceleration,

Â the A which is equal to the second derivative of this displacement,

Â the capital lambda, so A is equal to the capital lambda double prime.

Â So that will be equal to L times the theta double prime, that's the acceleration.

Â Then, divide the Newton's law of the motion to the force,

Â total force acting on the system is given by the N times A,

Â because A is equal to L times theta double prime,

Â the M times the A is equal to M times L times theta double prime,

Â which must be the same as the negative MG sine of theta,

Â that this right hand side is actually

Â the tangential component of the gravitational force W is equal to MG acting on the mass.

Â MG is acting to the mass in the direction opposite to the motion.

Â Let's look at the picture again.

Â Here you have a gravitational force,

Â W is a weight is equal to MG.

Â If this displacement angle is a theta then this angle is equal to theta.

Â So that we have another force,

Â tangential component of this gravitational force is MG times sine of theta,

Â that is this much.

Â But be careful that the direction of this,

Â the tangential component of the gravitational forces opposite to the motion.

Â So that F is equal to MA must to be equal to negative MG sine of theta.

Â That's the equation we get here.

Â Force to be compute,

Â the force F is equal to MA,

Â using this lambda is equal to L theta,

Â so that if we have M times L times and theta double prime,

Â then must be equal to negative of MG times A sine theta,

Â there is a tangential component of the gravitational force.

Â So, simplify this equation.

Â M is a common in both sides.

Â So divide us through M,

Â you are going to get theta double prime plus G of L and sine theta is equal to zero.

Â Second order, highly nonlinear differential equation.

Â When the absolute value of the angle is quite small,

Â then we know that sine theta can be approximated by a theta.

Â Because, from the calculus,

Â you know that limit X tends to zero of a sine X over X is equal to one.

Â And that means, sine X over X behaves like one for quite a small.

Â That's what I mean.

Â So if the absolute value of theta is quite small

Â than sine of theta is approximately theta.

Â So that instead of considering this nonlinear differential equation,

Â that's approximated by theta double prime plus G over L theta,

Â because a theta is a good approximation of sine of theta.

Â Now, this is the second order,

Â Constant coefficient homogeneous differential equation.

Â With the symbol Omega,

Â which is defined by the skillet of G over L,

Â you can be lighted as,

Â I said double prime plus Omega square times the theta is equal to zero.

Â That this equation, this a question,

Â we call it as a linearized equation of the original nonlinear differential equation.

Â Even though this is not the true governing equation,

Â but when the absolute value of theta is a quite small,

Â it will give us a good approximation of the pendulum motion.

Â And as you can see from this equation,

Â this is exactly the same as that differential equation.

Â Theta double prime plus omega skillet theta is equal to zero.

Â This differential equation is the same equation as

Â the equation for the spring master system for Pre- undamped Supreme Motion.

Â There is no external force and there is no damped pin.

Â And we know that the general solution of this differential equation is

Â theta is equals to C1 cosine Omega T plus

Â C2 sine Omega T. Which you can combine into

Â capital A times cosine omega T minus P.

Â With the capital P is skillet of C one squared plus a C2 squared.

Â That's the amplitude of this pendulum motion

Â and the P is to face angle and the C1 is given

Â by the Capital A times the cosine P and

Â the CT is equal to capital A times the sine of P, okay?

Â