# Work from the bottom of Unit circle to its top in Polar Coordinates

1. Oct 27, 2009

### soopo

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

Calculate the work $$W_{A B}$$ done by the force $$F$$ using Newton's laws ($$F=ma$$, etc), when a particle moves from the point $$A$$ to the point $$B$$ along the unit circle. The angle is $$\theta$$. No friction. How do you define kinetic energy in polar coordinates?

2. Relevant equations

Acceleration in polar coordinates is:

$$\bar{a} = ( \ddot{r} - r ( \dot{ \theta } )^2 ) \hat{r} + ( r \ddot{ \theta } + 2 \dot{ r } \dot{ \theta } ) \hat{ e_{\theta} }$$

3. The attempt at a solution

I know from cartesian coordinates that $$PE=KE <=> 1/2 * mv^2 = mg*h$$. I should verify it in polar coordinates. So integrating $$\bar{a}*m$$, with respect to the radius $$r$$ and the angle $$\theta$$, probably give me the energy, like $$W=F*distance$$ in carteesian coordinates.

Last edited: Oct 27, 2009
2. Oct 27, 2009

### jdwood983

The kinetic energy is given by $$T=\frac{1}{2}mv^2$$ and in polar coordinates, the velocity is given by

$$v(r,\theta)=\frac{dr}{dt}\hat{\mathbf{r}}+r\frac{d\theta}{dt}\hat{\boldsymbol{\theta}}$$

(you can check that taking the time-derivative of this does give the acceleration you have) so that takes care of the first part.

For the second part, you will need to integrate $$m\mathbf{a}$$ over $$r,\,\theta$$ to get the work:

$$\begin{array}{ll}W&=\int \mathbf{F}\cdot d\mathbf{r} \\ &=\int F_r\,dr+\int F_\theta\,rd\theta \\ &=\int ma_r\,dr+\int ma_\theta\,rd\theta \end{array}$$

3. Oct 27, 2009

### soopo

Why do you have $$r$$ in the last integral?

It seems that you have used the relation
$$dr = r d\theta$$

4. Oct 27, 2009

### jdwood983

It comes from the change of coordinate systems. If anything, the relation should be

$$d\mathbf{r}=dr\hat{\mathbf{r}}+rd\theta\hat{\mathbf{\theta}}$$

And when taking the dot-product with $$\mathbf{F}(r,\theta)$$, you get the integral relation I gave.

5. Oct 27, 2009

### soopo

How do you define $$KE$$ and $$PE$$ with respect to a conservative force $$\bar{D} = - m g \hat{k}$$ ?

Is it just a dot product? So I get the work, and then I can calculate the $$PE$$ with the new work:

$$\begin{array}{ll}W_{ n e w }&=\int \mathbf{D}\cdot d\mathbf{r} \\ &=0 \end{array}$$

Zero? I assumed $$\hat{k}$$ is perdendicular to the plane by $$\theta$$ and $$r$$. So $$PE$$ is zero with respect to the force $$\bar{D}$$.

Last edited: Oct 27, 2009
6. Oct 27, 2009

### jdwood983

I think you are combining two separate problems into one.

Work is defined as the force over a distance, and neither kinetic energy nor potential energy is a force, so you can't say that W=0 as you did above.

Kinetic energy is defined as the mass times the square of the velocity (really the square of the time derivative of the position). And potential energy is defined as the mass times the relative height times the acceleration due to gravity. Neither of these is work.

7. Oct 27, 2009

### jdwood983

(1) What is the work done by a force moving from point $$A$$ to $$B$$ in polar coordinates?
$$W=\int\mathbf{F}\cdot d\mathbf{r}=\int_A^BF_r\,dr+\int_0^{\phi}F_\theta\,rdr$$
where $$\phi$$ is the final angle, sweeping from $$A$$ to $$B$$. For the second problem,
$$KE=\frac{1}{2}mv^2=\frac{1}{2}m\left(\frac{dr}{dt}\hat{e}_r+r\frac{d\theta}{dt}\hat{e}_\theta}\right)$$