circular motion when an object travels about a given point at a set distance it is said to be in...
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Circular Motion
When an object travels about a given point at a set distance it is
said to be in circular motion
Cause of Circular Motion
– 1st Law…an object in motion stays in
motion, in a straight line, at a constant speed unless acted on by an outside force.
– 2nd Law…an outside force causes an object to accelerate…a= F/m
– THEREFORE, circular motion is caused by a force that causes an object to travel contrary to its inertial path
Circular Motion Analysisv1
v2
rr
Circular Motion Analysisv1
v2
rr0
v1
v2
v = v2 - v1
or v = v2 + (-v1)
(-v1) = the opposite of v1
v1
(-v1)
v1
v2
rr0 v = v2 - v1
or v = v2 + (-v1)
(-v1) = the opposite of v1
v1
(-v1)
v1
v2
v2
(-v1)
vNote how v is directed toward the center of thecircle
v1
v2
rr
v1
v2
v2
(-v1)
v
l
Because the two triangles aresimilar, the angles are equal andthe ratio of the sides areproportional
v1
v2
rr
v1
v2
v2
(-v1)
v
l
Therefore,
v/v ~ l/r and v = vl/r
now, if a = v/t and v = vl/r
then, a = vl/rt since v = l/t
THEN, a = v2/r
Centripetal Acceleration
ac = v2/r
now, v = d/t and, d = c = 2r
then, v = 2r/t and, ac = (2r/t)2/r
or, ac = 42 r2/t2/r or, ac = 42r/T2
The 2nd Law and Centripetal Acceleration
Fc
ac
vt
F = ma
ac = v2/r = 42r/T2
therefore,
Fc = mv2/r or,
Fc = m42r/T2
Motion in a Vertical Circle
A
B
Fw TA
Fw
TB
Vertical circle
Fw TA
Fw
TB
A
B
Top of Circle
at vmin TA = 0 and Fw = Fc
therefore, TA + mg = mv2/r
because TA = 0, mg = mv2/r
and v2 = rg
Vertical Circle
Fw TA
Fw
TB
A
B
Bottom of Circle
vmax at bottom
thereore, TB - mg = mv2/r
or Fc = TB - Fw
or TB = mv2/r + mg
Cornering on the HorizontalWhen an object is caused to travel in a circular path because of the force of friction,then,...
Fc = FF
car
Fw
FN
FF
Cornering on the Horizontal
Fc = FF
car
Fw
FN
FF
Therefore, mv2/r = FN
because FN = Fw = mg,
mv2/r = mg … or,
= v2 /rg
Cornering on a Banked Curve
car
Fw FN
Fp
Cornering on a Banked Curve
car
Fw FN
Fc
Fw FN
Fc
Note how FN isthe Resultant
Fw FN
Fc
If we want to know the anglethe curve has to be at to allowthe car to circle without friction,then we have to analyze theforces acting on the car.
Sin = Fc/FN and, Fc = SinFN