The tangential velocity of an object is directly proportional to the radius of the circular motion. How come orbital velocity is inversely proportional to the radius?
[Physics] Orbital Velocity and Tangential Velocity
orbital-motion
Related Solutions
Your understanding that the orbital velocity decreases as the radius increases is correct. Yet, as the article states, we see that orbiting stars seem to have a uniform speed. The resolution comes from the fact that the $M$ in the formula refers to the mass enclosed by the orbit (really the $M$ refers to a point mass, but an object a distance $R$ from the center of a spherically symmetric mass distribution feels the gravitational force that would result if all the mass inside the radius $R$ had been concentrated at the origin and the outside mass was removed completely).
For bigger orbits, $R$ is bigger as you noted, but also the orbits enclose more mass (mostly dark matter, as the article says), and so the orbital speed stays more or less constant. In fact it is precisely from this reasoning that the existence of dark matter was deduced.
Okay, regarding your first paragraph, I can't stop myself to say it again: this is only because English wants to be special. Most languages don't have a different word for "speed" and "velocity". This has been discussed before, but when you talk about force, or total force, or whatever, you can either mean $\vec{F}$ or $|\vec{F}|$, and nothing happens, so this is actually not relevant. I'll only use velocity here, please don't mind.
So, for any curvilinear movement, velocity is always tangent to the trajectory. You can give only $v$ in m/s, as you do know the direction; it is the curve's one.
In sum, the conversion orbital speed ↔ orbital velocity is immediate: just add/remove the unit vector tangent to the path.
As for tangential velocity, this is a more subtle issue. As I said, velocity is always tangential to the orbit, so it looks redundant to say "tangential velocity".
However, what it means is usually another different thing. Draw the planet and the center of forces (CoF). Of course the planet is moving around.
If you draw a radius from the CoF to the planet, you will have a "natural" axis for the planet. A perpendicular one completes a suitable natural reference frame.
This reference freme (local reference frame) varies with time, because it follows the movement of the planet.
The problem here is that notation is confusing. The "radial" component is fine (it's also called "normal component"), but how do we call the other one? It is usually called "tangential component" in English, but I don't like that word because it denotes it is tangent to the orbit, while it is not (not always).
I use to call it "transversal component". I find it less confusing.
This "transversal component" is only one of the components. There can be radial component too. Speed is the modulus of the vector, which you can find squaring both components and adding them up:
$$v=\sqrt{v_t^2+v_n^2}$$
Circular orbits are the only ones without radial velocity, so their velocity is both tangent to the orbit (as always) and purely transversal. However, any other orbit will have a radial component.
The key is being aware of the distinction between
- Tangential, in the sense that $\vec{v}$ is always tangent to the orbit; and this always happens, by definition.
and
- Tangential, in the sense of transversal, perpendicular to the radial component.
Edit:
Oh I forgot about rad/s. Obviously a so called "velocity" given in rad/s is obviously $\omega$. It's because of lazyness, but the correct name is angular velocity (or speed). Always check the units, that's what will tell you.
Best Answer
Orbital velocity generally is the pre-requisite velocity necessary for a satellite to stay in orbit. It is the minimum velocity necessary to keep the satellite from falling into the planet.
The shorter the radius of orbit, the closer the satellite is to the planet which it orbits, and the greater the gravitational attraction which must be overcome for it to remain in orbit. So a shorter radius means greater velocity required.
A longer radius puts the object farther out in space in weaker gravity and requires less velocity to keep it from falling into the planet. So a longer radius means less velocity required.
Tangential velocity is a result, not a pre-requisite. It is a function of the size of the circle the object describes and the number of rotations per unit time it makes. The larger the circle, the greater the distance which must be covered during each rotation. So if the object makes one rotation per minute, a longer radius means more distance covered during that minute, and greater velocity.
Greater orbital velocity results in more rotations per unit time. The orbital velocity is a requirement or pre-requisite for staying in orbit, whereas tangential velocity is a result of circular motion.
In other words, when you compute orbital velocity you specify the number of revolutions per unit time necessary to stay in orbit. This requirement becomes omega (rotations per unit time) in the tangential velocity formula.