Forces and their directions?

Question

I am finding a bit of confusion when it comes to equating forces.

For example :
When we consider satellites in a constant stationary orbit, we simply equate Newton's law of gravitation and the centripetal force equation, like so:

mwwr = Gmm/(RR)

Here, the centripetal force, acting downwards, is equal to the gravitational force, also acting downwards.
And I get that, they both have the same direction, so if the satellite stays in circular orbit, the forces must always be equal.

For my next example, when we equate Bqv with Eq in a velocity selector, we say that Bqv = Eq if the particle with charge q goes on in a straight line. (Due to a present crossed field)

The problem arises here.
If in gravitation, the gravitational force and the centripetal force act downwards both, I have no problem saying that they are equal.
However, in a crossed field, the electrostatic force is acting in the OPPOSITE direction of the magnetic force, so why is Bqv = Eq and not Bqv = -Eq?

Thanks I really need to have an answer for this, as I cannot understand this at all.

oldprof · Accepted Answer

OK, to start with "the centripetal force...is equal to the gravitational force" is wrong.  It's the gravitational force IS the centripetal force.  My point, the two forces are not equal, they are the same force.

And the gravitational force is the centripetal force in this case because it's turning the satellite inward, towards the center of rotation.  And that's the definition of centripetal force: any force that turns a body in motion towards its center of rotation.

So there's your direction...towards the center, it's in the name CENTRIpetal.

But gravitational force acts on all kinds of thing in all kinds of scenario.  So although it's the centripetal force on satellites, it might just be the weight of something when that something is not moving.  Your weight (watch the eclairs) is just the force of gravity with M the mass of Earth as the source of the force.  And this force is a pull towards the center of the mass M.

We define down by the direction of the pull of gravity.  So weight, W = mg is always...always...down; by definition.  But understand, this is the force of gravity; it's not the net weight or effective weight, which is the force of gravity modified by other forces, like centrifugal force.

Now let's look at F = qE + q(v X B) the EMF.

Note the cross product for a charge q moving with velocity v relative to the field B, the magnetic one.  As written, the magnetically induced EMF and the electrostatic EMF are aligned...pointing in the same direction.  If we simplify by assuming v crosses at 90 degrees, then F = qE + qvB is true.  So F = q(E + vB).

And there you are.  The two component forces are not...not...acting in opposite directions, given the assumptions.  But turn the charge around, so it crosses in the opposite direction, then we'd have F = q(E - vB) as you surmised.

My point, you need to specify which direction the charge is moving relative to the B field (or the B field moving relative to the charge).

So there you are.  If the two components are offsetting and the q is in orbit (like in an accelerator) it's because v is going the direction that gives us the F = q(E - vB) = 0 relationship.  In which case E = vB.  QED.

jeff · Answer

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Forces and their directions?

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