Pendulum Problem (Simpson's Rule)

[boogalu]

1. Consider a pendulum with length L that makes a maximum angle Θ with a vertical center line. The period T of this pendulum can be computed by the following integral
where k=sin(Θ/2) and g is the acceleration due to gravity. Find the period of this pendulum if L=1 meter and Θ=40º using Simpson's Rule with n=10

2.

3. So if I substitute 1 for L and 9.8 for g, I can multiple the integral by the constant in front of it. The problem I have is with the integral itself. Since k=sin(Θ/2), I can not see how to integrate with respect to x or Θ because they are both in the denominator of the equation. Can somebody please help me out with this by explaining what I should do ? I know the Simpson's rule formula but I am stumped how to apply it since I can not understand how to integrate with this formula ? All helpful advice is kindly appreciated !

 

[lanedance]

you only need to integrate w.r.t. x, theta is a constant in this equation as it represent the maximum angle, 40 deg for this case, so you can treat k as a constant

 

[boogalu]

what do you mean by integrate w.r.t. x ? I'm not familiar with that expression...

 

[gomunkul51]

w.r.t. x = with respect to x

 

[vela]

3. So if I substitute 1 for L and 9.8 for g, I can multiple the integral by the constant in front of it. The problem I have is with the integral itself. Since k=sin(Θ/2), I can not see how to integrate with respect to x or Θ because they are both in the denominator of the equation.

The presence of dx in the numerator tells you you're integrating with respect to x, not θ.

Can somebody please help me out with this by explaining what I should do ? I know the Simpson's rule formula but I am stumped how to apply it since I can not understand how to integrate with this formula ? All helpful advice is kindly appreciated !

Simpson's rule is a method for numerically evaluating an integral:

$$\int_a^b f(x)\,dx \cong \frac{h}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots+2f(x_{n-2})+4f(x_{n-1})+f(x_n)\right]$$

where n is the number of divisions, h=(b-a)/n, and x_i=a+ih. This problem is pretty much just plugging everything into the formula and grinding out an answer.

 

[boogalu]

thanks guys, I did as you said, split up the thing into the ten intervals and evaluated it at each, multiplied all of it by the gravity-pendulum constant in front and it came out to be 2.07~ seconds which the professor said was correct !