# Angular Momentum of a conical pendulum.

1. Jan 13, 2015

### Satvik Pandey

1. The problem statement, all variables and given/known data
A small ball of mass m suspended from a ceiling at a point O by a thread of length l moves along a horizontal circle with constant angular velocity $\omega$. Find the magnitude of increment of the vector of the ball's angular momentum relative to point O picked up during half of revolution.

2. Relevant equations

3. The attempt at a solution

I
nitial velocity of ball $V_{i}=v\hat { j }$

Initial distance of the ball from O is (R)=$lsin\alpha\hat{i}-lcos\alpha\hat{k}$

Final velocity $V_{f}=-V\hat{j}$

Final distance of the ball from O is $-lsin\alpha\hat{i}-lcos\alpha\hat{k}$

Initial momentum is $R$ cross $P$.

$L_{i}=mvLcos\alpha\hat{i}-mvlsin\alpha\hat{k}$

$L_{f}=-mvLcos\alpha\hat{i}+mvlsin\alpha\hat{k}$

$\delta L=-2mvLcos\alpha\hat{i}+2mvlsin\alpha\hat{k}$

So its magnitude is $2mvL$

I got $cos\alpha=\frac{g}{\omega^2l}$ by writing the force equation.

Now $v=lsin\alpha\omega$

Using this I got the answer

$2ml^{2}\omega\sqrt{1-\frac{g^{2}}{(\omega^{2}l)^2}}$

2. Jan 13, 2015

### Satvik Pandey

Can you guys tell me some applications like Daum Equation Editor for writing equations in LaTex. I don't why is it not working in my PC.

3. Jan 13, 2015

### ehild

Check your coordinate system and the components of your vectors, they do not match

4. Jan 13, 2015

### Satvik Pandey

I have considered the circle to be in X-Y plane.

I have shown the top view also.

For finding angular momentum of any body about any point(say O) we drop any line from O to the line of the velocity of that body. Then we calculate $\overrightarrow { R } \times \overrightarrow { p }$. right?

So $R_{i}=lsin(\alpha)\hat { i }- lcos(\alpha)\hat { k }$

5. Jan 13, 2015

### ehild

OK, but then the change the top right figure with the unit vectors. And check the signs in Li .
I got

$L_{i}=mvLcos\alpha\hat{i}+mvlsin\alpha\hat{k}$

6. Jan 13, 2015

### Satvik Pandey

$\frac { 2mgl }{ \omega } \sqrt { 1-{ \left( \frac { g }{ { \omega }^{ 2 }l } \right) }^{ 2 } }$.