Central force and acceleration in the polar direction

In summary: It means that the magnitude of the velocity in the theta direction (and in all other directions, for that matter) is not constant.
  • #1
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Consider a central force. The central force is radial by definition, so ##\vec{F}=f(r) \hat{r}##. Therefore, by definition, the acceleration caused by the force, in the direction of ##\hat{\theta}## must be zero, ##\vec{a_{\theta}}=0##.

In presence of central force angular momentum is conserved, in particular its magnitude, which is ##|\vec{L}|=m r^2 v_{\theta}##, where ##v_{\theta}## is the magnitude of the velocity of the body in the direction of ##\hat{\theta}##.
That means that the product ##r^2 v_{\theta}## stays constant and, if ##r## gets smaller, ##v_{\theta}## must increase.

Since ##v_{\theta}## is the magnitude of the velocity of the body in the direction of ##\hat{\theta}##, this means that there must be and acceleration in this direction, necessarily.

But how can this be possible, since, as said, the force is completely radial?

I'm ok with the fact that the force is not orthogonal to the velocity, hence the magnitude of the velocity can increase, but, in particular, I really don't see how the magnitude of the component ##v_{\theta}## can change, provided the fact that the force is radial.

In this picture it is clear that the magnitude of the velocity (which is tangential to the ellipse, of course) changes, but I can't understand the change in ##v_{\theta}##.

https://www.physicsforums.com/attachments/99029
 
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  • #2
No, it doesn't mean that there is a theta component. It means that the initial velocity is v_theta. If there was an acceleration along theta, angular momentum wouldn't be conserved. There would be a non-zero net torque.
 
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  • #3
What does v equal in polar coordinates?
 
  • #4
Soren4 said:
I'm ok with the fact that the force is not orthogonal to the velocity, hence the magnitude of the velocity can increase, but, in particular, I really don't see how the magnitude of the component ##v_{\theta}## can change, provided the fact that the force is radial.
Before you take on elliptical orbits in a central force, consider the simpler case of no force at all. What is the equation of motion of a particle that starts at the point ##(\theta=0,r=1))## with a speed ##v## in the tangential direction? It's not subject to any force or acceleration at all, yet neither the ##\theta## nor the ##r## components of its velocity are constant.

Cartesian coordinates have the nice property that the basis vectors are functions of neither position nor time, so when you rewrite ##\vec{F}=m\vec{a}## as the differential equation ##\frac{d^2\vec{r}(t)}{dt^2}=\frac{\vec{F}}{m}## it simplifies into ##\frac{d^2A_x(t)}{dt^2}=\frac{F_x}{m}## (and likewise for the y and z components) and you can safely conclude that if ##F_x=0## then ##v_x## will be a constant. More general coordinate systems don't necessarily work that way.
 
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  • #5
vθ not being constant does not mean that the acceleration in the theta direction is not zero.
 
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1. What is a central force in the polar direction?

A central force in the polar direction is a type of force that acts towards or away from a central point, following a polar coordinate system. This means that the force is directed along a radial direction, which is defined by the distance from the central point and the angle with respect to a fixed reference direction.

2. How is acceleration related to central force in the polar direction?

Acceleration is directly related to central force in the polar direction. According to Newton's Second Law of Motion, the acceleration of an object is equal to the net force acting on the object divided by its mass. In the case of central force in the polar direction, the acceleration will always be directed towards or away from the central point, depending on the direction of the force.

3. What are some examples of central forces in the polar direction?

Some common examples of central forces in the polar direction include gravity, electric and magnetic forces, and even the force exerted by a spring. These forces all follow a polar coordinate system, with the force being directed along a radial direction towards or away from a central point.

4. How is the magnitude of central force in the polar direction determined?

The magnitude of central force in the polar direction can be determined by using the formula F = ma, where F is the force, m is the mass of the object, and a is the acceleration. Since the force and acceleration are both directed along the same radial direction, the magnitude of the force will be equal to the magnitude of the acceleration multiplied by the mass of the object.

5. What is the relationship between central force in the polar direction and circular motion?

Central force in the polar direction is closely related to circular motion. In fact, any object moving in a circular path is experiencing a central force in the polar direction, which is directed towards the center of the circle. This force is responsible for keeping the object moving in a circular path and is known as the centripetal force.

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