CIRCULAR MOTION
Basic Rotational Quantities
The angular displancment is defined by:
3600 = 2prad
1 rad = 57.30
1rev = 3600
Rev to Rad to Degree
(2)
Circular Motion
For circular motion at a constant speed v, the centripetal
acceleration of the motion can be derived. Since in radian
measure, (2)
Notice that the velocity vector (in
blue) is constantly changing direction.
Even though the magnitude (amount)
of the velocity stays constant, the
direction is changing. This shows that
the object is accelerating. Remember
that acceleration is defined as the
rate of change of velocity. So any
change, even if it is just the direction,
is an acceleration. Another way to
think about this is to consider that to
change your direction of motion
requires a net force, and a net force
causes acceleration. (1)
Centripetal Acceleration
The centripetal acceleration expression is
obtained from analysis of constant speed
circular motion by the use of similar triangles.
From the ratio of the sides of the triangles (2)
Centripetal Acceleration
ac = Centripetal acceleration
SI: m/s2
vT = Tangential velocity or speed SI: m/s
r = Radius of object's path
SI: m
w = Angular velocity
SI: rad/s
In order for an object to execute circular
motion - even at a constant speed - the object
must be accelerating towards the center of
rotation
Centripetal Force
Any motion in a curved path represents accelerated motion, and
requires a force directed toward the center of curvature of the
path. This force is called the centripetal force which means
"center seeking" force. The force has the magnitude (2)
The centripetal
acceleration can be
derived for the case
of circular motion
since the curved path
at any point can be
extended to a circle
Note that the centripetal
force is proportional to the
square of the velocity, implying
that a doubling of speed will
require four times the
centripetal force to keep the
motion in a circle. If the
centripetal force must be
provided by friction alone on a
curve, an increase in speed
could lead to an unexpected
skid if friction is insufficient
(2)
Centripetal Force Calculation (2)
Centripetal force = mass x velocity2 = mac
radius
The smaller the mass, the smaller the centripetal
force (shown by the red vector labeled as the
force of tension in the rope, FT) you will have to
apply to the rope (1)
The smaller the velocity of the object, the less centripetal force you will
have to apply. (1)
Notice that the
centripetal force
and the centripetal
acceleration are
always pointing in
the same direction
The smaller the length of rope (radius), the more centripetal
force you will have to apply to the rope.
If you let go of the rope (or the rope breaks) the
object will no longer be kept in that circular path
and it will be free to fly off on a tangent. (1)
It is conceptually better to think about the Centripetal force that is
calculated from the formula as a requirement. If you meet the
requirement, then you have circular motion at the radius and speed used
in the formula. If you do not meet the requirement, then the object
moves into a larger curve (which requires less force) or defaults into
straight line motion (going off on a Tangent). (1)
In this animation, the
"sticky" or adhesive
forces from the mud to
the tire tread are large
enough to be the
centripetal force
required to keep the mud
in a circular path as the
tire spins. (1)
In this animation, the tire is
spinning faster which means a
larger centripetal force is
required to keep the mud in
the circular path of the
tire. The adhesive forces of
the mud to the tire are not
large enough to meet the
requirement. The mud begins
to move into a larger circular
path but as soon as it is not
touching the tread then there
is no force (other than
gravity) and so the mud
continues with the velocity it
had at the instant it was no
longer touching the tread. (It
went off on a tangent). It
followed Newton's first law!
(1)
Centrifugal Force - The False Force
Centrifugal force does not exist...there is
no such thing...it is a ghost we tend to blame
odd behavior on.
Take for example this
common situation. You
are riding in a car going
around a curve. Sitting
on your dashboard is a
cassette tape. As you go
around the curve, the
tape moves to outside
edge of the car.
The car tires on the road have a enough static friction to act as
centripetal force which forces the car to go around the curve. The
tape on the slippery dashboard does not have enough friction to act as a
centripetal force, so in the absence of a centripetal force the tape
follows straight line motion. (1)
Centripetal Force Example
The string must provide the necessary centripetal force to move the
ball in a circle. If the string breaks, the ball will move off in a straight
line. The straight line motion in the absence of the constraining force is
an example of Newton's first law. The example here presumes that no
other net forces are acting, such as horizontal motion on a frictionless
surface. The vertical circle is more involved. (2)
Problem 1:
A 150g ball at the end of a string is swinging in a horizontal circle
Of radius 0.60m. The ball makes exactly 2.00 revolutions in a second
What is its centripetal acceleration?
Problem 2:
The moon’s nearly circular orbit about the earth has a radius of about
385,000km and a period of 27.3 days. Determine the acceleration
Of the moon toward the earth.
V = 2pr
T
Problem 3:
A 1000kg car rounds a curve on a flat road of radius 50m at a speed
Of 50km/h. Will the car make the turn is (a) the pavement is dry and
The coefficient of static friction is 0.60 (b) the pavement is icy and
The coefficient of friction is 0.20
On a bank: FNsinq = m v2
r
1. http://regentsprep.org/Regents/physics/phys06/bcentrif/default.htm
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2. http://hyperphysics.phy-astr.gsu.edu/hbase/cf.html