Download Document

Key Concepts for the Lecture of 21Jul03
Circular Motion
&
The position of a particle traveling around a circular path of radius r at constant speed v can
be given in three ways:
1. By the usual x and y coordinates . (For this discussion we will take the origin at the center
of the circular path). The vector r = xi + yj is called the radius vector.
2. By the path length s traversed since passing some arbitrary point (which we will take to be
the point where the x-axis crosses the circle).
3. By the angle that a radius vector makes with the x-axis (measured counterclockwise
from the x-axis).
Note that
s r
x r cos y r sin tan
&
&
1
y
x
Angle is a ratio of two lengths ( = s / r ) so it is unitless. As a convenience we refer to angle
as having units of radians, saying, for example, that the angle subtended by half a circle is Œ
radians, but this is a notational convenience. Angle is dimensionless.
Speed is the rate of change of s; that is v = ds/dt. The rate of change of the angular position is called the angular speed
& d
dt
The angular velocity is a vector with magnitude & and direction perpendicular to the plane of
rotation. The direction is found using the right hand rule. Curl the fingers of your right hand
in the direction of motion and your thumb will point in the direction of &. Counterclockwise
is out of the plane; clockwise is into the plane. Increasing is out of the plane.
Note that
v ds
d
r
r&
dt
dt
&
Angular velocity is a ration of an angle to a time ( & = û / ût ), so it has units of radians per
second. But the radian is actually dimensionless, so properly, the units of angular velocity
are inverse seconds, or sec-1.
&
A particle following a circular path, even at constant speed, is constantly changing the
direction of its velocity. It is therefore accelerating. The only effect of the acceleration is to
change the direction of the velocity vector.
&
Since the acceleration of a particle in uniform circular motion serves only to change the
direction of the velocity but not the speed, the acceleration vector must always be at right
angles to the velocity. The acceleration vector has no component in the direction of travel.
The velocity vector is always tangent to the circular path, so the acceleration always points
toward the center of the circle, at a right angle to the velocity. Since this acceleration is
always pointing towards the center of the circle, it is called centripetal acceleration.
&
The centripetal acceleration of the particle is
aC v2
r &2
r
To derive this expression, consider two velocity vectors corresponding to two moments in
time separated by a small time increment ût. The difference between these two vectors is the
change in velocity ûv over the time increment ût. During this time increment, the particle’s
angular position has changed an amount û = rût. Consequently the length of ûv is
ûv v û
The direction of ûv is towards the center of the circle.
The magnitude of the acceleration is a = ûv/ût, so the acceleration is toward the center with
magnitude
aC ûv
û
v
v&
ût
ût
&
v2
r &2
r
According to Newton’s first law, a particle left to itself will move in a straight line at
constant speed; that is, it will have constant velocity. A particle moving in a circle is
accelerating, so according to Newton’s second law there must be some force causing it’s
centripetal acceleration. In order for a particle to follow a circular path, something must
be pushing or pulling on the particle to make it go in a circle. A force is required to supply
the centripetal acceleration. This force, whatever its source, is called the centripetal force.
According to Newton’s second law, the centripetal force is
F3C m a3C
The agency providing the centripetal force must arrange that the force is always directed
toward the center of the circle (at right angles to the instantaneous velocity) with a magnitude
FC m a C m
v2
m r &2
r
&
Centripetal force causes circular motion; circular motion does not cause centripetal force.
Centripetal force is just the name we give to any force that is exerted on a particle at a right
angle to its velocity.
&
All forces arise from interactions between pairs of bodies. In order to have a centripetal force
on a particle, there must be some other object pushing on it. According to Newton’s third
law, the object pushing on the particle undergoing circular motion must feel a reaction force.
That reaction force is sometimes called the centrifugal force. Centripetal means “center
seeking”; centrifugal means “center fleeing.”
&
When a centripetal force stops acting, the particle that was going in a circle continues in a
straight line tangent to the circle. Something must keep supplying a centripetal force in order
to keep a particle bending its path into a circle.