# Question about central force,and acceleration of the tangent velocity due to a radial velocity and angular momentum conservation

I am a bit confused, and I'll appreciate some help.

Let's say that a particle is moving under the influence of a central force. Let's say that its initial velocity, has both: tangent component and radial component.

If I am not mistaking, the radial component of the velocity, will cause the radius to become smaller. If the radius become smaller, then from the conservation of angular momentum dictated by the central force, the component of the tangent velocity, supposed to become bigger.

But, central force does not perform work, at least it cannot increase the magnitude of the tangent velocity. So who is increasing the magnitude of the tangent velocity?

What am I missing here?

• @ AL_P : the picture you have in mind is like of a planet moving around the Sun? Anyway, what about the centrifugal force? As you describe the radial force, it seems a centripetal force, like a gravitational attraction. Is my analogy correct? Well, but it is not the only one acting on the arena. Dec 6 '14 at 13:07

A central force does not perform work only if the motion is tangential. When the particle moves radially, it has a component of the speed that is not tangent. The cetripetal force and the velocity are no longer paralell so there the centripetal force actually does work on the system. The work is actually $W=\int_{r_i}^{r_f}F_{centripeta}(r) dr$

NOTE: for circular motion you usually use a non-inetrial rotating system where the object is at rest. This introduces a radial pseudoforce, the cetrifugal force. But if the system of reference does not move with the object, such as when there is radial motion, you need to take into account two additional pseudoforces.

One is the Euler force: the fictitious tangential force that is felt in reaction to any radial acceleration.

The other is the Coriolis force, which creates a deflection of moving objects when the motion is described relative to a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right.

• Thanks for the reply. I understand what you are saying, but my confusion is about the growing velocity in the direction perpendicular to the radius, when the radius becomes shorter (conservation of angular momentum) though the central force has no component in this direction. I assume that my confusion is related to my attempt to look at the problem from Newton's laws perspective although this system is not inertial, but a rotating one. Thanks again.
– AL_P
Dec 12 '14 at 8:34
• @AL_P you are correct, there are additional non-radial pseudoforces when you are in a non inertial reference frame. They only have an effect when the object is moving in that rotating refernce frame. I updated my answer to clarify it more.
– user65081
Dec 12 '14 at 16:20