# How did they get this form

1. Aug 18, 2009

### AntiStrange

1. The problem statement, all variables and given/known data
I'm supposed to read this little packet on mechanics, specifically dealing with central forces.
and they are explaining a scenario where something is moving due to a central force (say, the earth rotating around the sun) where all the motion takes place in the xy plane. They describe some initial conditions y0 = z0 = 0 for example and Vz = 0 because it's not moving in the z direction.
Then they switch to polar coordinates (r,theta) and they say the equations of motion are:
$$m\ddot{r} - mr\dot{\theta}^2 = F(r)$$
$$mr\ddot{\theta} + 2m\dot{r}\dot{\theta} = 0$$

2. Relevant equations
Force: F = m*a
Angular Velocity: $$\omega = \frac{v_{t}}{r}$$
Angular Acceleration: $$\alpha = \frac{d\omega}{dt}$$
Centripetal Acceleration: $$\alpha_{c} = \omega^{2}r = \frac{v^{2}_{t}}{r}$$

v_t = tangential speed

3. The attempt at a solution
well for the first equation of motion they give, I recognize that the first part is a force m*a because $\ddot{r}$ is an acceleration. Because it's the acceleration of the radius it should be the centripetal force right? But I don't understand why there needs to be another force component subtracted from this one, because the only force acting on this object should be the central force. In addition to that, the second component appears to ALSO be the same central force because $\dot{\theta}$ is the angular velocity $\omega$ and that is squared times the radius which is just another way to find the centripetal acceleration. So I don't understand why there needs to both, let alone why they must be subtracted.

edit: the two dots aren't showing up above the theta below, but instead of (theta)r it should be d^2(theta)/dt^2 * r
For the second equation of motion they have $\ddot{\theta}r$ is just the angular acceleration times the mass would be any force that acts tangential to the motion, which should not exist in this case which is why it is equal to zero. But again I don't know why they include a second part.

2. Aug 18, 2009

### kuruman

The first equation is Newton's Second Law in the radial direction. Imagine an object spiraling out moving faster and faster as time increases. In the radial direction, the force does two things: (a) it causes the component of the velocity vr to increase as the object spirals out (first term) and (b) it causes the object to go around a curved path (second "centripetal" term).

If you consider an object going around a circle, i.e. r = constant,

$$\dot{r} = 0$$

$$- mr\dot{\theta}^2 = F(r)$$