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Uniform Circular Motion
UCM
, a special case of 2-D motion where:
-^
The radius of motion (r) is constant
-^
The speed (v) is constant
-^
The velocity changes as object continually changesdirection
-^
Since
is changing (v is not but direction is), the object
must be accelerated:
-^
The direction of the acceleration vector is always toward thecenter of the circular motion and is referred to as thecentripetal acceleration
-^
The magnitude of centripetal acceleration:
ˆ^
yˆ
x
c^
x^
y
v
v
a =a
a^
x +
y
t^
t
^
^
^
^
^
v
r
2
c^
c
v
a =a =
r
v
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Derivation of Centrifugal Acceleration
Consider an object in uniform circular motion, where:
r = constant {radius of travel}v = constant {speed of object} The magnitude of the centripetal acceleration can be
obtained by comparing the similar triangles for positionand velocity:
x v r y
x
y
The direction of position and velocity vectors both shift by thesame angle, resulting in the following relation:
2 c
r^
v^
v^
r^
v^
v^
r
=
v=
=
v a =
r^
v^
r^
r^
t^
r
t
^
^
^
^
^
^
r^1
r^2
r Position only
v v v
Velocity only
r 2 v time, t
2
time, t v1^ r^1
1
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Centripetal Force & Gravitation
Assume the Earth travels about the Sun in UCM:Questions:
1.^
What force is responsible for maintain the Earth’s circular orbit?
2.^
Draw a Free-Body Diagram for the Earth in orbit.
3.^
Calculate gravitational force exerted on the Earth due to the Sun.
4.^
What is the centripetal acceleration of the Earth?
5.^
What is the Earth’s tangential velocity in its orbit around the Sun?
Sun
Earth
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Centripetal Force vs. Centrifugal Force
•^
An object moving in uniform circular motion experiences anoutward “force” called centrifugal force (due to the object’sinertia)
-^
However, to refer to this as a force is technically incorrect.More accurately, centrifugal force _______.– is a consequence of the observer located in an accelerated (revolving)
reference frame and is therefore a fictitious force (i.e. it is not really aforce it is only perceived as one)
- is the perceived response of the object’s inertia resisting the circular
motion
(& its rotating environment)
- has a magnitude equal to the centripetal force acting on the body
Centripetal force & centrifugal force are
NOT action-reaction pairs…
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Banked Curves
-^
Engineers design banked curves to increasethe speed certain corners can be navigated
-^
Banking corners decreases the dependenceof static friction (between tires & road)necessary for turning the vehicle into acorner.
-^
Banked corners can improve safetyparticularly in inclement weather (icy or wetroads) where static friction can becompromised
-^
How does banking work?
-^
The normal force (F
) of the bank on the carN
assists it around the corner!
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Vertical Circular Motion
•^
Consider the forces acting on a motorcycle performing aloop-to-loop: • Can you think of other objects that undergo similarmotions?
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