The Pendulum Equation In this project, you will develop and solve the pendulum e

Question

The Pendulum Equation In this project, you will develop and solve the pendulum equation, which is a classic differential eqa tion problem We'll start with a pendulum of length To clarify, we are going to find the equation of motion of , the angle of displacernent of the pendulum from its equilbirum (the red dashed line). 1. First, we will look at the forces exerted on the pendulum. We'll use F ma. First, develop a formula for the tangential force, F in terms of the angle of displacement, . To begin, note that Fa mg Use that, and trigonometry, to find an expression for F. 2. Next, we'll deal with the acceleration, the a in F = ma. It follows first that the position of the tip of the pendulum as travels is given by 0. Recalling that acceleration is the second derivative of position, find an expression for a. 3. You should now have F and a, and m is a known constant. Write F = ma using what you found in parts (1) and ( Put everything on the left hand side of the equation and ta-da!, you have a differential equation! Now, this differential equation that you have is actually not in our toolbox to solve. Instead, we have to approximate the sin ) term using the first term of the MacLaurin series approximation for sin ) What is the MacLaurin series approximation for sin(0)? 5. Replace sin(0) with the first term in the MacLaurin series for sin(), what you should have now is a linear, second order, constant coefficient problem

Explanation / Answer

1] Tangential force F = FG sin theta

= mg sin theta

  

2] a = s'' here '' represents double differentiation with respect to time

= (l theta)''

3] Here F = ma

a = g sin theta

s'' = g sin theta

(l theta)'' = g sin theta

l * theta '' = g sin theta

4] first term of mclaurin expansion of sin theta is theta

so sin theta = theta

5] l * theta '' = g theta where l is length

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