Friday, September 23, 2011

Uniform circular motion describes an object that is travelling at constant speed in a circular pathway. Though the speed which is the magnitude of the velocity remains the same, the direction of velocity is constantly changing as the object curves along the edge of the circle. A changing velocity implies a nonzero acceleration, and thus a net force acting on the object. What might such an acceleration or force look like?

The equation which characterizes UCM is

||Fnet|| = mar = mv2/r

where m is the mass of the object, v the speed, and r the radius of the pathway.  The acceleration ar = v2/r and net force point at all times toward the center of the circle. You might realize this is exactly Newton’s law with the added information that the acceleration is entirely radial.

Why would that be? If this were true, it means the external forces on the object in sum continually change the direction of an object’s motion but preserve its speed. This means the object never gains nor loses kinetic energy! It also means that acceleration, pointing radially toward the center of the circle, is always at odds to the velocity, which points along a tangent to the circle. By geometry the acceleration and velocity vectors for a particle in UCM are always perpendicular, because in a circle a radial line and a tangent must form a right angle at the circumference.

We gain more insight by looking to Newton’s laws. The first defines inertia, which is the ability of an object with mass to remain at the same velocity unless acted upon by a force. So we know that, absent this curious radial net force, the object would speed through space not only at a constant speed but in a straight line. With the forces in place, a net acceleration arises that snaps the trajectory from line to circle.

The second law states ∑ F = ma, that the acceleration of the object is result of all the forces acting on it. UCM is usually defined using this equation, so there’s not much to glean here except to remember that the characteristic acceleration, v2/r, is constant in time because v and r are constant, and that we must sum all the forces before equating to mv2/r. But this is a sticking point: the net force in UCM is often called centripetal force, but it is not actually present in the setup. When the initial conditions of the system are just right that they effect a constant radial acceleration, only then do we call the net force centripetal in nature. The UCM version of Newton’s second law conveniently allows us to solve for forces or other unknowns precisely because we know what the resultant net force must be.

Finally, Newton’s third law states for any force from a first object on a second, the force from the second on the first is equal in magnitude and opposite in direction. This is useful for looking at specific situations: celebrated (or not) problems include cars travelling on a circular track, cars travelling on a circular banked track, the carnival ride where a person is pressed against the inside of a spinning cylinder, a dish being carried on a lazy susan, a puck or other object being spun around on a string, or an electron shot through some kinds of magnetic or electric fields. Forces between objects can be clues to weigh in the equation that equates to the centripetal (net) force.

If you have some calculus under your belt, here is a fun exercise: imagine a bicycle chain, which is a closed loop made of metal links. You set the chain spinning so that it continues spinning in a perfect circle. If the chain has n links, what is the tension in each link? Then what is the tension in each link as n approaches infinity?

If you’ve been following the blog, try formulating UCM using the Frenet-Serret equations!

### Notes

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sweet lord jesus this is just genious *-*
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