# Classical mechanics question (pendulum)

1. Nov 8, 2017

### Clara Chung

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
I have done part a, I have no idea on part b, here is my attempt,

2. Nov 8, 2017

### 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.

3. Nov 8, 2017

### Clara Chung

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)

4. Nov 8, 2017

### 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 $x$.

5. Nov 8, 2017

### Clara Chung

(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} ?

6. Nov 8, 2017

### TSny

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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7. Nov 9, 2017

### TSny

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.

8. Nov 9, 2017

### Clara Chung

Still ∅ is θ0

9. Nov 9, 2017

### vela

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

10. Nov 9, 2017

### TSny

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

11. Nov 9, 2017

### Clara Chung

Theta o only

12. Nov 9, 2017

### Clara Chung

Ahhh I understand it should be θ0 / 2