Circular Motion
Chapter 7.3
What moves in a circle?
•
•
•
•
Day 1
The earth around the sun
A car on a curve
A disk on a string
A tetherball
Why does it move in a circle?
• The earth around the sun
– Gravity
• A car on a curve
– Friction
• A disk on a string
– The tension in the string
• A tetherball
– The tension in the string
Centripetal Force
• We call any force that keeps something moving in
a circle a centripetal force
• In order to keep something moving in a circle, we
must provide a force to counter the tendency of
inertia
• What does the inertia of the object make it want to
do?
Centripetal Acceleration
• To keep something moving in a circle we must
constantly provide a force
• The direction of the velocity is constantly
changing
– What do we call a change in velocity (a vector quantity)
with time?
• Is a body moving in a circle at equilibrium?
• Is there a net unbalanced force acting on the
object? (Is there an acceleration?)
Centripetal Force
• Centripetal force is affected by:
– The mass of the object (m).
– The speed of the object around the circle (v).
– The radius of the circle (r).
• Using Newton’s 2nd Law of Motion (F = ma),
centripetal force is mathematically represented as
follows:
Fc = mac = mv2
r
Day 1
ac = v2
r
Motion & Forces
• What you already know:
– Velocity – a measure of the change in displacement
(distance with direction.
– Mass – A measure of the amount of matter an object
contains.
– Acceleration – A measure of the change in velocity
over change in time.
– Force – A push or pull that is equal to the mass of the
object multiplied by its acceleration (F = ma).
Uniform Circular Motion
• Uniform circular motion is defined by any object
that is moving at constant speed in a circular path.
– Determining Speed:
» The distance an object moving in a circular path is
equal to the circumference (C = 2r).
» The time it takes an object to complete one
revolution is called the period (T).
» It then follows that the speed of an object moving in
a circular path can be determined by:
v = d/t = C/T = 2r/T
Uniform Circular Motion
• If an object is moving at constant speed in a
circular path, can it be accelerating?
– Yes
» Although the speed may be constant, the direction is
changing.
» If direction is changing over time, then the velocity
must be changing.
» Acceleration is the change in velocity over time
(a = v/t).
» If the velocity is changing over time, then the object
must be accelerating.
Circular Motion – Instantaneous
Velocity
• Note that the velocity vector is at right angles to the
position vector and tangent to the circle at any given point
along the circle.
r2

v2
r1
r2

r1
r
v = r/t
v1
Instantaneous v (arrived at by
letting  get smaller and smaller)
is tangent to the circle.
Circular Motion – Centripetal
Acceleration (ac)
• The acceleration of an object moving in a circular path
always points towards the center of the circle, and is
perpendicular to the velocity vector.
v2
v
v2
v
a
r

v1
v1
a = v/ t
Centripetal Acceleration
• The angle between r1 and r2 is the same as the
angle between v1 and v2.
– Therefore, the triangles these vectors make are similar
such that:
r/r = v/v
– If you divide both sides by t:
r/(t r) = v/(t·v)
– Since :
» r/t = v and v/t = a
– Hence:
v/r = a/v
and
ac = v2/r
Centripetal Acceleration
• An alternative representation for centripetal
acceleration can be derived using the
circumference and period of revolution.
» d = 2πr
» v = d/T = 2πr/T
– Substituting into ac = v2/r
» ac = (2πr/T)2/r
» ac = 4π2r/T2
Circular Motion – Centripetal Force
• To make an object move in a circular path, an external
force must act perpendicular or at right angles to its
direction of motion.
• This force is called centripetal force.
Instantaneous
direction of velocity
Direction of force
required to make
object move in a
circular path
(towards the center)
Centripetal Force
• Centripetal force is affected by:
– The mass of the object (m).
– The speed of the object around the circle (v).
– The radius of the circle (r).
• Using Newton’s 2nd Law of Motion (F = ma),
centripetal force is mathematically represented as
follows:
F = mv2
r
Note: Centripetal force is an unbalanced “net” force
How the Factors Affect
Centripetal Motion
• Which graph shows the proper
relationship with respect to force:
– Force vs. Mass.
Speed
– Force vs. Speed.
Radius
– Force vs. Radius.
Mass
Objects that travel in circular paths.
What is the cause of the force?
• The Earth – Sun System:
– Gravity.
• A racecar traveling around a turn on the
racetrack:
– Friction.
• An athlete throwing the hammer:
– Tension in the cable attached to the hammer.
The path of objects.
• If the centripetal force were suddenly removed
from an object moving in a circular path, what
trajectory (or path) would it follow?
Which Path?
Why?
• Because of Inertia – An object in motion wants to remain
in motion at constant speed in a straight line.
• If the unbalanced centripetal force is removed, the object
will continue in a straight path.
(a)
(b)
(c)
Example #1:
• A 1.5 kg cart moves in a circular path of 1.3
meter radius at a constant speed of 2.0 m/s.
– Determine the magnitude of the centripetal
acceleration. (ac = v2/r )
– Determine the magnitude of the centripetal
force. (Fc = mac )
– Determine the period. (T = d/v = c/v)
– Get out your calculators!
Example #1: (cont.)
• Centripetal Acceleration:
ac = v2/r = (2.0 m/s)2/(1.3 m) = 3.1 m/s2
• Centripetal Force:
Fc = mac = (1.5 kg)(3.1 m/s2) = 4.6 N
• Period:
T = c/v = 2r/v = 2(1.3 m)/(2.0 m/s) = 4.08 s
Example #2: Roller Coaster
m = 1kg
r = 10m
v = 9.9m/s
ac = ??m/s^2
Fc = ??N
The centripetal force is the force keeping the object in a
circle. It is either going to be Fg, FN, or some combination.
Here Fg = Fc. Fg IS
Fc. There is no
Normal force. You
feel weightless.
Fg = mg = Fc
Fc = N
Fg = mg
Here Fc = FN and points to the center.
Fg points straight down but if you are
accelerating with it you only feel Fc.
(And feel confused if you have time.)
Fg = mg
Fg = mg
Here Fg = Fc but point in
opposite directions. The
Normal force has to oppose
Fg and provide Fc. You feel
2 “g”s.
Example #2: Roller Coaster
The track of the roller coaster provides the
centripetal force via the Normal force, FN
Example #3: Ball on a rope
A tetherball of mass 0.28kg is moving at 8m/sec
around a pole. (For this example, we will ignore
gravity.) The length of the rope is 2.4m.
What is the period of motion?
What is the centripetal force?
Ignoring gravity, what is the tension in the rope?
(For this example, we will ignore gravity.)
Example #3: Ball on a rope
Period T = d/v = 2Pr/v = 2P(2.4m)/(8m/s) = 1.9s
(This is not actually needed to solve the problem.)
What is the centripetal force?
Fc = mac = mv2/r = 0.28kg * (4m/s) 2/2.4m
What is the tension in the rope?
The tension = Fc
Example #4: Car on a curve
A 1000kg car is traveling at 22m/s on a curve of radius 100m.
What provides the centripetal force required to keep the car
from sliding off the road?
What is the magnitude of the centripetal force required to keep
the car on the road?
What is the coefficient of friction between the car and the
road?
F = mv2/r = 1000kg *(22m/s)2/100m = 4840N.
Example #4: Car on a curve
What provides the centripetal force required to keep the car
from sliding off the road? Friction
What is the magnitude of the centripetal force required to keep
the car on the road?
Fc = mv2/r = 1000kg *(22m/s)2/100m = 4840N.
What is the coefficient of friction between the car and the
road?
m = Ff/FN = Fc/FN = Fc/(mg) = 4840N/(1000kg * 9.81m/s2) =
0.5
Example #5: Moon around the Earth
The moon has a mass of 7.35 × 1022 kg and orbits at an
average distance of 3.84 × 108 m in 27.32 days (2.36 ×
106sec).
What centripetal acceleration does the moon experience and
what is the required centripetal force?
ac = 0.0027m/s2
Fc = mac = (7.35 × 1022 kg)(0.0027m/s2) = 1.98 × 1020N