FORCES AND CIRCULAR MOTION:
The important thing to understand about this subject is that there is no new
force involved. It may seem like a new force-- go on some of those
amusement-park rides and experience those non-forces for yourself and you will
decide that this guy is loony. No, i speak the truth.
Read on. First, think about an elevator ride. You are standing in the elevator,
and there are two forces on you, the gravitational force (mg) down and the
normal force (N or FN) up. (N and FN are the two commonly
used sy
When the elevator accelerates, N and mg are the forces on you, but the N adjusts itself so that the vector sum of the forces is in the direction of the acceleration and is equal to ma. N is very smart.
A similar situation is a ball hanging on a string. There is the force mg
down and tension T up. They are equal and opposite if there is no acceleration.
Now whirl it around in a circle. No new force acts, but T adjusts itself so
that the net force is in the direction of acceleration and is equal to ma. T is
very smart. Now to be as smart as T, all you need to know is that the component
of acceleration toward the center of the circle is
v2/r.
(Why is the acceleration v2/r?) Now apply S f = ma and we find that the sum of all components of force toward the center = mv2/r.
When you ride a roller coaster and you are at the top or bottom of a hill, two vertical forces act on you: mg and N. Weight mg acts toward the center of Earth and N acts perpendicular to and away from the track. N very cleverly adjusts itself so that the vector sum of the two forces is toward the center of curvature and is equal to mv2/r. This is true whether you are upside-down at the top of a loop or right-side up at the top or bottom. If feels funny because you are so accustomed to the normal force in everyday life being equal and opposite to mg.
Note that i did not mention centripetal or
centrifugal force. These are downright misleading terms. When you get to the
essence of what is going on, you will not need to mention them. People use
these words in order to simplify a discussion of circular motion or to discuss
it without getting into the details, but then the truth of what is really
happening tends to get hidden. A good rule of thu
Here are some examples:
Below is a car coming toward you on a banked turn going just the right speed
such that there is no friction force perpendicular to the direction of motion.
There is a normal force on each wheel, and the FN shown is the sum
of the four. If the banking angle is q
from horizontal, FN is tilted q
from vertical. If the turn is in a horizontal plane, choose a horizontal x axis
and make it positive to the left, so the car has positive acceleration to the
left (toward the center of curvature). The write the equations and solve for v.
Don't peek.
S Fy
= 0 or FN cosq - mg = 0, and
S Fx
= ma or FN sinq = mv2/r.
Solve the first for FN and plug it into the second, and you will
find that m cancels out, showing that big trucks and little cars should travel
the same speed on the turn. Also do you know that sinq /cosq = tanq ?
Below is a ball on a string, traveling in a vertical circle. The
gravitational force mg has a component mgcosq toward the center, and tension T is toward
the center so
T + mgcosq =
mv2/r. The ball could be traveling in either direction. This could
be a roller coaster on one of those loops; then replace T with the normal force
N or FN.
A pendulum is similar to the above except that it does not go all the way around, just back and forth at the bottom:
Note that q in this picture is not
the same as in the previous picture.
S Fc
= mv2/r, so T - mgcosq = mv2/r.
In the direction of motion S f = ma
becomes
mgsinq = ma.
Back. to the pasture, the mechanics stuff, or maybe you came from the Newton's 2nd law page or from the derivation of ac = v2/r.
Other main pages:
Fluids, heat, electricity and magnetism
Vibrations and waves
Quantum
or all kinds of stuff alphabetically in my index.
F. Gram,
Comments, questions: fredrick.gram at tri-c.edu (but remove
“at” and spaces and insert @)