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Chapter 7
Circular Motion
Chapter Objectives
•Relate radians to degrees
•Calculate angular quantities such as displacement,
velocity, & acceleration
•Differentiate between centripetal, centrifugal, &
tangential acceleration.
•Identify the force responsible for circular motion.
•Apply Newton’s universal law of gravitation to find the
gravitational force between two masses.
Parts of the Circle
Angle
measured in
radians (Θ)
Arc length (s)
Reference
line
Radius (r)
Radians v Degrees
Radians are another way to measure an angle. We
use radians in circular motion because it is a unitless number as opposed to degrees. That way it
carries no units to mix up our final measurements.
You will often see radians measured with π in it.
That is what we want since π is a real number with
no units.
Converting Between Radians and
Degrees
To convert from degrees
to radians, simply multiply
Θ(degrees) x π/180
Remember that radians involve π, so we want the degrees
to disappear and leave the π.
To convert from radians to degrees, do the inverse of
above
Θ(radians) x 180/ π
We now want the radians to go away, so that means π must be
divided out.
Angular Displacement
Angular displacement is the distance an object
travels along the circumference of a circle. This is
used to measure the speed of a orbiting satellite or a
rock tied to the end of a string.
ΔΘ =
Angular displacement
(radians)
Δs
change in
arc length
r
radius
Angular Velocity
Angular speed is defined much like linear speed in
which the displacement of the object is measured for
a specific time interval.
omega
ω=
ΔΘ
Δt
angular
displacement
angular speed
(radians/second)
or revolutions per
time
time
Angular Acceleration
While we are on the same path as linear motion,
we can use linear acceleration to formulate an
equation for angular acceleration.
α=
Δω
Δt
angular velocity
(rad/s)
angular
acceleration
(rad/s2)
Time (s)
Rotational Kinematics
One Dimensional
Rotational
v = v0 + a Δt
ω = ω0 + αΔt
Δx = 1/2(v + v0) Δt
v2 = v02 + 2aΔx
Δx = v)Δt + 1/2aΔt2
ΔΘ = ½(ω + ω0)Δt
ω2 = ω02 + 2αΔΘ
ΔΘ = ω0Δt + ½αΔt
Tangential v Centripetal
• Tangential follows the
guidelines of linear
quantities.
• So tangential speed is
the instantaneous
linear speed of an
object traveling in a
circle.
• Tangential acceleration
is the instantaneous
linear acceleration of
an object traveling in a
circle.
•
Centripetal is a term
associated with circular
motion.
• Centripetal means
center-seeking.
• Centrifugal means
center-fleeing.
Tangential Speed
Tangential speed is the thought that as an object is
traveling in a circle, with what speed is it traveling
linearly.
Or a more practical use would be if the object were to
break its circular motion, what path would it travel?
So what would the initial velocity be of the object as
it breaks from the circle? Linear
ω
ΔΘ = Δs
r
Δt
Δt
radius
velocity
arc length
Now solve for velocity by multiplying
both sides by r.
vt = rω
This
equation
only works
when ω is
in radians
per unit
time.
Tangential Acceleration
Tangential acceleration is again that instant where the
circular motion breaks and linear motion takes over.
So basically we are converting from circular to linear
motion.
And remember that acceleration is just the rate of
change of velocity.
vt = rω
tangential
acceleration
Δt
Δt
Rate of change means divide
by time.
angular
acceleration
at = rα
Centripetal v Centrifugal
• Remember that
centripetal means center
seeking.
• And centrifugal
means center fleeing.
Acceleration in a Circle?
•
Recall that acceleration can occur in two ways
1.
2.
The magnitude of the velocity changes.
The direction of the velocity changes.
And since the
instantaneous
•
Now
will
we swung
call it centripetal or centrifugal
Imagine
a rock
being
velocity at
acceleration
based
on
its
direction?
on a string in a circular path.
those two
points run
tangential to
Since acceleration is found
the circle, we
by the change of velocity,
can draw
we must have two different
vectors to
represent the
velocities and two different
two different
times.
velocities and
two different
times.
Zoom In a Little
We have found two different velocities at two different times so
we can find the acceleration.
But we want to know the acceleration the instant the string would
break, that way we can use our tangential velocity concept.
So we have found
two velocities of the
rock at the two
times as close
together as
possible.
And now recall the
formula for acceleration
is finding the difference
of the velocities over
the time it took to
change the velocity.
v
v0
v – v0
Δv
=
a=
Δt
Δt
Subtracting Vectors Graphically
Remember to place them
head-to-head.
And the order is important to find
the resultant, so draw the
resultant from the final to the
initial.
v
-v0
Δv
And notice the change in velocity points toward the center. So the
acceleration is seeking the center. So we call this
Centripetal Acceleration
Formula for Centripetal
Acceleration
vt2
ac = r
ac =
rω2
Use if you are given a
tangential velocity. Usually
identified by a unit of distance
over time.
Use if you are given angular
velocity. That angular velocity
must be in radians per time.
Total Acceleration
• The total acceleration takes the tangential
acceleration and the centripetal acceleration into
account at the same time.
• That is because the tangential acceleration
takes into account the changing speed and the
centripetal acceleration takes into account the
changing direction.
So,
at = √(at2 + ac2)
Centripetal Force
• Since acceleration is centripetal, the force must also be
centripetal because it follows the direction of the
acceleration.
• So centripetal force is the force responsible for
maintaining circular motion.
• The reason you feel a force pulling out is because
inertia is resisting the centripetal force of circular motion.
Formula for Centripetal Force
We derived our universal formula for force from Newton’s 2nd Law.
F = ma
Using a little substitution of the formulas for centripetal
acceleration.
vt2
vt2
F=m r
ac = r
ac = rω2
F = m rω2
Newton’s Universal Law of
Gravitation
• Isaac Newton observed that planets are held in
their orbits by a gravitational pull to the Sun and
the other planets in the Solar System.
• He went on to conclude that there is a mutual
gravitational force between all particles of
matter.
• From that he saw that the attractive force was
universal to all objects based on their mass and
the distance they are apart from each other.
• Because of its universal nature, there is a
constant of universal gravitation for all
objects. G = 6.673 x 10-11 Nm2/kg2
Formula for Newton’s Universal
Law of Gravitation
Fg = G
Force due to gravity.
Same concept that we
have seen before.
m1m2
r2
Constant
of
Universal
Gravitation
Distance
between
the centers
of mass of
the two
objects.
Masses of the
two objects.
Acceleration Due to Gravity
• We have seen this before, but from this Universal Law of
Gravitation, we can calculate the acceleration due to
gravity.
• You simply treat one of the masses as the mass of the
Earth, and the distance between objects becomes the
radius of the Earth.
Fg = G
m1m2
=
2
r
MEm2
G R2 =
E
ME = 5.98 x 1024 kg
RE = 6.37 x 106 m
ME m
GR
E
ag