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Analogies between linear and rotational motion, two types of circular motion, centripetal, and centrifugal forces, weightlessness, common situations involving centripetal acceleration.
2.
Circular motion is very
similar to linear motion in
many ways.
Linear Angular
Quantity unit Quantity unit
Displacement (x) M Angular Rad
Velocity (v) m/s displacement (ϴ)
Acceleration (a) m/s/s Angular velocity (ω) Rad/sec
Mass (m) Kg Angular Rad/sec/sec
acceleration (α)
Force (F) Kg * m/s or N
Moment of inertia Kg * m*m
(ɪ )
Torque (t) N*m
3.
Analogies Between
Linear and Rotational
Motion
4.
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.
5.
Centripetal acceleration –
acceleration of an object in
circular motion. It is directed
toward the center of the circular
path.
2 ac = centripetal acceleration,
v m/s2
ac =
r v = tangential speed, m/s
r = radius, m
6.
Centripetal Force – the net inward
force that maintains the circular
motion of an object. It is directed
toward the center.
Fc = centripetal force, N
Fc = m a c m = mass, kg
2
mv ac = centripetal
Fc = acceleration,
r m/s2
v = tangential speed, m/s
r = radius, m
7.
Two types of speed
Linear speed is the distance traveled per unit of
• 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.
8.
Tangential Speed (linear)
Tangential speed depends on two things
1. rotational speed
2. the distance from the axis of rotation.
9.
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.***
10.
Rotational speed (sometimes called angular
speed) is the number of rotations per unit of
• 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).
11.
Rotational Speed
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)
12.
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.
13.
Question # 1
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?
Her rotational speed is also 4 RPM, and her linear
speed is 3 m/s.
14.
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
15.
For a constant tangential speed:
v = tangential speed, m/s
d 2πr d = distance, m
v= =
t T t = time, s
r = radius, m
T = period, s (time for 1
rev.)
If rpm (revolutions per minute) is given, convert to m/s using
these conversion factors:
1rev 2 r and 1 min. = 60 sec.
Or you can find the period by taking the inverse of the
1 T = period, s – time for one revolution
T= F = frequency, rev/s – number of revolutions per time
f Note: Period and frequency are inverses.
16.
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.
17.
BIG IDEA….
Centripetal force keeps an
object in circular motion.
18.
Centripetal Force
The force exerted on a whirling can is toward the center. NO
outward force acts on the can.
19.
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)
20.
Creates a curved path
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.
21.
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)
Orbital motion, such as a satellite: Fc = Fg
(weight or force of gravity)
22.
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)
Orbital motion, such as a satellite: Fc = Fg
(weight or force of gravity)
A person going around in a spinning
circular room: Fc = FN (normal force)
23.
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)
Orbital motion, such as a satellite: Fc = Fg
(weight or force of gravity)
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)
24.
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
25.
Example :Motion in a
Vertical
v
Circle
v
v Consider the forces on a ball attached to
v +
Bottom
Tmg a string as it moves in a vertical loop.
+
T T Note changes as you click the mouse
T
+ mg to show new positions.
+
mg + mg
T The velocity of the object is constantly
changes depending on which direction
v gravity is pointing compared to
Top
Left of Path
Side velocity.
Top
Top mg
RightRight
Tension is minimum +
The tension required to keep this
as weight
Weight
Maximum has helps FcT, W
no effect
tension object moving in a circle changes
Weight causes small
force
on T
Weight
opposes has
Fcin no effect while it is in it motion as well.
decrease tension T
on T Bottom
27.
Calculating Centripetal Forces
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.
28.
Adding Force Vectors
•A conical pendulum is a bob held in a circular path by a string attached
•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.
29.
•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.
30.
Centripetal Force
Centripetal force keeps the vehicle in a circular path as it rounds a
banked curve.
31.
Centrifugal Forces – MISCONCEPTION!!
•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.
32.
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
33.
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
34.
Satellites and “Weightlessness”
The satellite is kept in orbit by its speed—it is
continually falling, but the Earth curves from
underneath it.
35.
Satellites and “Weightlessness”
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
36.
Satellites and “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:
37.
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
38.
Level Curves
Friction is the
force that
produces the
centripetal
acceleration
Can find the
frictional force, µ,
or v
v rg
39.
Banked Curves
A component of
the normal force
adds to the
frictional force to
allow higher
speeds
v2
tan
rg
or ac g tan
40.
Vertical Circle
Look at the forces
at the top of the
circle
The minimum
speed at the top of
the circle can be
found
v top gR
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