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Chapter 7
Circular Motion
Chapter 7 Outline
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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.
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.
Section 7.1
Measuring Rotational
Motion
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
vf = vi + a Δt
ωf = ωi + αΔt
Δx = 1/2(vf + vi) Δt
ΔΘ = ½(ωf + ωi)Δt
vf2 = vi2 + 2aΔx
ωf2 = ωi2 + 2αΔΘ
Δx = viΔt + 1/2aΔt2
ΔΘ = ωiΔt + ½αΔt2
Section 7.2
Tangential and Centripetal
Acceleration
Tangential v Centripetal
• Tangential follows the
• Centripetal is a term
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.
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? Linear
So what would the initial velocity be of the object as it
breaks from the circle?
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
Section 7.3
Causes of Circular Motion
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. The magnitude of the velocity changes.
2. The direction of the velocity changes.
Now will we call it centripetal or centrifugal acceleration based
on its direction?
Imagine a rock being swung
on a string in a circular path.
Since acceleration is found
by the change of velocity,
we must have two different
velocities and two different
times.
And since the
instantaneous velocity
at those two points run
tangential to the circle,
we can draw vectors to
represent the two
different 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.
vf
vi
And now recall the formula for
acceleration is finding the
difference of the velocities over
the time it took to change the
velocity.
vf - vi
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.
vf
vi
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
FC = m r
ac = r
ac = rω2
FC = 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.