# What does centripetal force affect?

If an object is moving in a circular path with a particular radius and velocity, you can calculate the centripetal force acting on that object. If you change the centripetal force (lets say you increase it), does it change the velocity or radius? or both? and by how much?

What I'm asking pertains to magnetism. If a particle with charge

"q" is traveling at velocity "V" in the 'x' direction and passes through a magnetic field "B" in the 'z' direction which is perpendicular to its path it will experience a force "F" acting in the 'y' direction. The force changes the direction in which the particle is traveling, so now the direction of the force changes, which changes the path again, yadda yadda yadda!.... the particle is moving in a circular path and the force acting on it is centripetal... that is all understood. And since you know the velocity, the mass, and the force, you can determine the radius of that circular path.

Now! if you increase the strength of the magnetic field "B", you therefore increase the strength of the centripetal force "F". As this force increases, does it affect only the radius (making it smaller) and velocity remains constant? Or, does an increase in centripetal force increase only velocity and radius remains constant? Or both? and by how much?

Thank you!

Eric

Update:

In regards to Veritas De Vita:

So you are saying that ONLY the radius changes, not the velocity/speed?

because that makes sense, going back to F=qVB... Increase B to B' and force increases to F'... and IF "r" is somehow held constant, then V HAS to increase to V' to equal that new centripetal force F'. but if you plug that new V' back into F=qVB with your new B' that you changed originally, then your F' is not the same as when you first increased B to B' like it should, and F' itself will increase AGAIN to F'' and therefore so will your V... AGAIN! and the process infinitely repeats. which just doesn't make sense.

In other words, V CANNOT change or the equation wont work, therefore ONLY r can change... because in that system of magnetism there is no way to maintain a constant r, is there? and if the V DOES change, it screws up the whole equation, making it impossible to work backwards and determine what you changed originally, VorB...

Am I on the right track? or is my theory wrong?

Relevance

Hi Eric,

This is actually a very good question! I spent quite a while trying to understand what was going on here.

Basically, it boils down to this (in magnetism :P):

In magnetism, the force is always perpendicular to velocity of the particle in the field. As a result, the speed of the particle is never changed.

---MATHEMATICAL EXPLANATION---

You can see this if you use vectors as follows (It's hard to show vectors on here - but I'll try my best!):

If v is the velocity and a is the acceleration (both are vectors), then you can summarise the fact that the force (F) is perpendicular to the velocity by

v.F = 0, where the dot product is used, and hence

v.a = 0 (since F = ma)

Integrating this, you get 1/2 v.v = constant, so v.v = constant, so speed = |v| = sqrt(v.v) = constant.

Thus the speed is constant when the particle is solely in a magnetic field.

---END---

Thus, whether you increase the magnetic field strength or not, the particle will always have the same speed, since the force is always perpendicular to the velocity of the particle.

The centripetal force on the particle due to the magnetic field is equal to mv²/r (or qv x B from the magnetic equations, where 'x' is the cross product). Thus if the force is increased by a factor of A, then since m and v are constant, you can see that the radius is decreased by a factor of A.

Let me know if any of this doesn't make sense!

Btw the case is different for something like a gravitational field. In that case, if you have a particle in a circular orbit and suddenly increase the centripetal force on it, then the particle will find itself moving in an ellipse instead of a circle.

Hope this helps :)

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I think what you've written sort of makes sense... Re-writing what you're saying, I think you could write it as follows:

The centripetal force F = qvB = mv²/r, where q and m are constant.

Assume r is constant. Then qB = mv/r, so B = (mq/r)v, which implies that the magnetic field strength B is proportional to the speed of the particle v. This is incorrect, and so we can deduce that our assumption was wrong, and r is not constant.

However, I don't think an argument like the one you have can let you deduce that v is constant, because if you do repeat the process infinitely, like you've suggested, you may find that v converges to a particular value.

I don't know if you've done any vectors, so the explanation I gave earlier for the speed staying the same might not have made much sense. However, I'm afraid I can't think of a much better way of describing it... The basic principle is that because the magnetic force is always perpendicular to the velocity of the particle, it only changes the direction, and never the speed - and this is true whenever a force is at right angles to the motion of a particle, for example, in gravity.

(Be careful when you say the velocity doesn't change - the velocity does change, because the velocity is speed in a particular direction, and the direction does change - it's just that the speed does not)

So whenever a particle is solely in the influence of a magnetic field, it doesn't lose speed at all. (Having said that, I think there is some stuff about how accelerating charged particles are supposed to emit light - but you won't ever need to worry about that until quite late in a university course! I'm not even sure if that would make the speed change, it might just make the particle spiral inwards...)

Please e-mail me if you are at all unsure about any of this! :)

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• If mass increased, centripetal force increases because F=ma. If radius increased, centripetal force decreases because centripetal acceleration, a.k.a. radial acceleration is a=v^2/r. Plug that into F=ma, and you get F=(mv^2)/r. If tangential velocity increased, centripetal force would increase.

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• Fc = Mass* (tangential velocity)^2 / radius

Fc = m*(V^2/r)

You can change any of the factors or any combination of factors such that the equation balances.

You have:

Fc

m

v

r

Tell me any 3 and I will tell you the fourth

If you add another constraint, like conservation of energy and hold mass and velocity constant. ====> then you must adjust r to compensate for a change in Fc.

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