Circular_Motion 2019.ppt

Circular motion is very similar
to linear motion in many
ways.
Linear
Quantity unit
Displacement (x) M
Velocity (v) m/s
Acceleration (a) m/s/s
Mass (m) Kg
Force (F) Kg * m/s or N
Angular
Quantity unit
Angular displacement
(ϴ)
Rad
Angular velocity (ω) Rad/sec
Angular acceleration
(α)
Rad/sec/sec
Moment of inertia (ɪ) Kg * m*m
Torque (t) N*m

Analogies Between Linear and
Rotational Motion

There are two types of circular
motion
An axis is the straight line around
which rotation takes place.
• When an object turns about an internal
axis—that is, an axis located within the
body of the object—the motion is called
rotation, or spin.
• When an object turns about an external
axis, the motion is called revolution.

Centripetal acceleration –
acceleration of an object in circular
motion. It is directed toward the
center of the circular path.
2
c
v
a =
r
ac = centripetal acceleration, m/s2
v = tangential speed, m/s
r = radius, m

Centripetal Force – the net inward force
that maintains the circular motion of an
object. It is directed toward the center.
c c
2
c
F = m a
m v
F =
r

Fc = centripetal force, N
m = mass, kg
ac = centripetal acceleration,
m/s2
r = radius, m

Linear speed is the distance traveled per unit of
time.
• A point on the outer edge of the turntable travels
a greater distance in one rotation than a point
near the center.
• The linear speed is greater on the outer edge of a
rotating object than it is closer to the axis.
• The speed of something moving along a circular
path can be called tangential speed because
the direction of motion is always tangent to the
circle.
Two types of speed

Tangential speed depends on two things
1. rotational speed
2. the distance from the axis of rotation.
Tangential Speed (linear)

Which part of the turntable moves faster—the outer part where the
ladybug sits or a part near the orange center?
***It depends on whether you are talking about linear speed or
rotational speed.***
Faster?

Rotational speed (sometimes called angular
speed) is the number of rotations per unit of
time.
• All parts of the rigid turntable rotate about
the axis in the same amount of time.
• All parts have the same rate of rotation, or
the same number of rotations per unit of
time.
• It is common to express rotational speed
in revolutions per minute (RPM).

All parts of the turntable rotate at the
same rotational speed.
• A point farther away from the center travels a longer path
in the same time and therefore has a greater tangential
speed. (Linear Speed)
Rotational Speed

Remember All parts of the turntable rotate at the same rotational
speed.
A point farther away from the center travels a longer path in the
same time and therefore has a greater tangential speed.(linear
speed)
Therefore, a ladybug sitting twice as far from the center
moves twice as fast.
Rotational Speed

At an amusement park, you and a friend sit on a large
rotating disk. You sit at the edge and have a rotational speed
of 4 RPM and a linear speed of 6 m/s. Your friend sits
halfway to the center. What is her rotational speed? What is
her linear speed?
Answer:
Her rotational speed is also 4 RPM, and her linear speed is 3
m/s.
Question # 1

Calculating Average Speed
 An object moving in uniform circular motion would cover
the same linear distance in each second of time.
 When moving in a circle, an object travels a distance
around the perimeter of the circle.
 The distance of one complete cycle around the perimeter
of a circle is known as the circumference.
The circumference of any circle is
Circumference = 2*pi*Radius

For a constant tangential speed:
d 2 π r
v = =
t T
d = distance, m
t = time, s
r = radius, m
T = period, s (time for 1 rev.)
1
T=
f
T = period, s – time for one revolution
F = frequency, rev/s – number of revolutions per time
Note: Period and frequency are inverses.
If rpm (revolutions per minute) is given, convert to m/s using these
conversion factors: and 1 min. = 60 sec.
Or you can find the period by taking the inverse of the frequency.
1 2
rev r



Constant Speed, but is there
constant Velocity?
 Remember speed is a scalar quantity
and velocity is a vector quantity.
 The direction of the velocity vector is
directed in the same direction that the
object moves. Since an object is
moving in a circle, its direction is
continuously changing.
 The best word that can be used to describe
the direction of the velocity vector is the
word tangential.

BIG IDEA….
Centripetal force keeps
an object in circular
motion.

The force exerted on a whirling can is toward the center. NO
outward force acts on the can.
Centripetal Force

Since centripetal force is a net force, there
must be a force causing it. Some examples are
 A car going around a curve on a flat road: Fc =
Ff (friction force)

Centripetal force holds a car in a curved path.
a. For the car to go around a curve, there must be sufficient friction to
provide the required centripetal force.
b. If the force of friction is not great enough, skidding occurs.
Creates a curved path

Ff (friction force)
 Orbital motion, such as a satellite: Fc = Fg
(weight or force of gravity)

Ff (friction force)
 A person going around in a spinning circular
room: Fc = FN (normal force)

Ff (friction force)
 A person going around in a spinning circular
room: Fc = FN (normal force)
 A mass on a string (horizontal circle, i.e.. parallel
to the ground): Fc = T (tension in the string)

For a mass on a string moving in a vertical circle,
the centripetal force is due to different forces in
different locations.
 At the top of the circle, Fc = T + Fg (tension plus
weight or gravity)
 At the bottom of the circle, Fc = T - Fg (tension
minus weight or gravity)
 On the outermost side, Fc = T
 Anywhere other than above, you would need to
find the component of gravity parallel to the
tension and either add or subtract from tension
depending on the location on the circular path

Example :Motion in a
Vertical Circle
Consider the forces on a ball attached to a
string as it moves in a vertical loop.
Note changes as you click the mouse to
show new positions.
The velocity of the object is constantly
changes depending on which direction
gravity is pointing compared to velocity.
The tension required to keep this object
moving in a circle changes while it is in it
motion as well.
+
T
mg
v
Bottom
Maximum tension T, W
opposes Fc
+
v
T
mg
Top Right
Weight has no effect on
T
+
T
mg
v
Top Right
Weight causes small
decrease in tension T
v
T
mg
+
Left Side
Weight has no effect on
T
+
T
mg
v
Bottom
v
T
mg
Top of Path
Tension is minimum as
weight helps Fc force
+

Greater speed and greater mass require greater centripetal force.
Traveling in a circular path with a smaller radius of curvature requires a
greater centripetal force.
Centripetal force, Fc, is measured in newtons when m is expressed in
kilograms, v in meters/second, and r in meters.
Calculating Centripetal Forces

•A conical pendulum is a bob held in a circular path by a string attached
above.
•This arrangement is called a conical pendulum because the string sweeps
out a cone.
•Only two forces act on the bob: mg, the force due to gravity, and T,
tension in the string.
• Both are vectors.
Adding Force Vectors

•The vector T can be resolved into two perpendicular components,
Tx (horizontal), and Ty (vertical).
•Therefore Ty must be equal and opposite to mg.
•Tx is the net force on the bob–the centripetal force. Its magnitude
is mv2/r, where r is the radius of the circular path.
Remember…

Centripetal force keeps the vehicle in a circular path as it rounds a
banked curve.
Centripetal Force

•When an object moves in a circular motion there MUST be an
outward force.
•NO!!!
•This apparent outward force on a rotating or revolving body is
called centrifugal force. Centrifugal means “center-fleeing,” or
“away from the center.”
•If there was an outward force, we would see something
completely different than what actually happens.
Centrifugal Forces – MISCONCEPTION!!

Gravity Near the Earth’s Surface
The acceleration due
to gravity varies over
the Earth’s surface
due to altitude, local
geology, and the
shape of the Earth,
which is not quite
spherical.

Satellites and “Weightlessness”
Satellites are routinely put into orbit around the
Earth. The tangential speed must be high enough
so that the satellite does not return to Earth, but
not so high that it escapes Earth’s gravity
altogether.

The satellite is kept in orbit by its speed—it is
continually falling, but the Earth curves from
underneath it.

Objects in orbit are said to experience weightlessness. They
do have a gravitational force acting on them, though!
The satellite and all its contents are in free fall, so there is no
normal force. This is what leads to the experience of
weightlessness.

More properly, this effect is called apparent
weightlessness, because the gravitational force
still exists. It can be experienced on Earth as
well, but only briefly:

Common situations involving Centripetal Acceleration
 Many specific situations will use forces
that cause centripetal acceleration
 Level curves
 Banked curves
 Horizontal circles
 Vertical circles
 Note that Fc, v or ac may not be constant

Level Curves
 Friction is the force
that produces the
centripetal
acceleration
 Can find the
frictional force, µ, or
v
rg
v 


Banked Curves
 A component of the
normal force adds
to the frictional
force to allow
higher speeds
2
tan
tan
c
v
rg
or a g





Vertical Circle
 Look at the forces at
the top of the circle
 The minimum speed
at the top of the circle
can be found
gR
vtop 

Circular_Motion 2019.ppt

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Circular_Motion 2019.ppt