Classical mechanics question (pendulum)

[Clara Chung]

Homework Statement

Homework Equations

The Attempt at a Solution

I have done part a, I have no idea on part b, here is my attempt,

 

[kuruman]

This is all very confusing. What is $\phi$? The radical in the first line of your development should be $\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}$. Also, the rest of the stuff in the integrand doesn't look right either. Please show your steps in more detail.

 

[Clara Chung]

This is all very confusing. What is $\phi$? The radical in the first line of your development should be $\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}$. Also, the rest of the stuff in the integrand doesn't look right either. Please show your steps in more detail.

Ummm phi is theta o ...and I changed the integral in part a to the integral in the attempt by substituting x = sin(theta/2) / sin ( theta o)
Then dx = cos (theta/2) / 2sin(theta o) d(theta)

 

[TSny]

I think you're OK so far (after realizing that $\phi = \theta_0$). In the expression $\frac{dx}{\cos \left(\theta / 2 \right)}$, express $\cos \left(\theta / 2 \right)$ in terms of $x$.

 

[Clara Chung]

I think you're OK so far (after realizing that $\phi = \theta_0$). In the expression $\frac{dx}{\cos \left(\theta / 2 \right)}$, express $\cos \left(\theta / 2 \right)$ in terms of $x$.

(I am using a as theta and b as theta o because I can't type them)
dx/cos(a/2) = dx / √(1-sin^2(x)sin^2(b))
So its approximation is
dx / {1-sin^2(x)sin^2(b)/2} ?

 

[TSny]

(I am using a as theta and b as theta o because I can't type them)
dx/cos(a/2) = dx / √(1-sin^2(x)sin^2(b))
So its approximation is
dx / {1-sin^2(x)sin^2(b)/2} ?

EDIT: Did you mean to have (b/2) as the argument in sin²(b)?

You can continue to simplify this using the fact that b is small.

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[TSny]

(I am using a as theta and b as theta o because I can't type them)
dx/cos(a/2) = dx / √(1-sin^2(x)sin^2(b))

Another approach is to use the small angle approximation directly on $\frac{1}{\cos \left( \theta /2 \right)}$ rather than first expressing $\cos \left(\theta /2 \right)$ in terms of $\sin \left(\theta /2 \right)$. But your method will work also with about the same amount of effort.

 

[Clara Chung]

EDIT: Did you mean to have (b/2) as the argument in sin²(b)?

You can continue to simplify this using the fact that b is small.

You can enter Greek letters by using the tool bar. Click on Σ.
View attachment 214652

There are also buttons on the tool bar for superscript and subscript.

Still ∅ is θ₀

 

[vela]

When you used the trig identity to rewrite the integrand in terms of sine, what happened to the factor of 2 multiplying $\sin^2$?

 

[TSny]

In post #8, should $\phi$ stand for $\theta_0$ or $\theta_0 / 2$?

 

[Clara Chung]

In post #8, should $\phi$ stand for $\theta_0$ or $\theta_0 / 2$?

Theta o only

 

[Clara Chung]

Theta o only

Ahhh I understand it should be θ_{0 / 2}