Transcript Circular Motion - wiscoscience.com

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 Analysis
v1
v2
r
q
r
Circular Motion Analysis
v1
v1
v2
r
0
r
v2
Dv = v2 - v1
or Dv = v2 + (-v1)
(-v1) = the opposite of v1
v1
(-v1)
v1
v1
v2
r
r
0
v2
Dv = v2 - v1
or Dv = v2 + (-v1)
(-v1) = the opposite of v1
v1
Dv
v2
(-v1)
(-v1)
Note how Dv is directed
toward the center of the
circle
v1
v1
Dl
r
q
v2
r
v2
Dv
v2
q
(-v1)
Because the two triangles are
similar, the angles are equal and
the ratio of the sides are
proportional
v1
v1
Dl
r
q
v2
v2
r
Dv
v2
q
(-v1)
Therefore,
Dv/v ~ Dl/r
and Dv = vDl/r
now, if a = Dv/t and Dv = vDl/r
then, a = vDl/rt since v = Dl/t
THEN, a = v2/r
Centripetal Acceleration
ac = v2/r
now, v = d/t
and,
d = c = 2pr
then, v = 2pr/t
and,
ac = (2pr/t)2/r
or, ac = 4p2 r2/t2/r
or,
ac = 4p2r/T2
The 2nd Law and Centripetal
Acceleration
vt
Fc
F = ma
ac
ac = v2/r = 4p2r/T2
therefore,
Fc = mv2/r
or,
Fc = m4p2r/T2
Motion in a Vertical Circle
A
Fw
TA
TB
B
Fw
Vertical circle
A
Top of Circle
at vmin TA = 0
Fw
TA
TB
B
Fw
and Fw = Fc
therefore, TA + mg = mv2/r
because TA = 0, mg = mv2/r
and
v2 = rg
Vertical Circle
A
Bottom of Circle
vmax at bottom
Fw
TA
TB
B
Fw
thereore, TB - mg = mv2/r
or Fc = TB - Fw
or
TB = mv2/r + mg
Cornering on the Horizontal
When an object is caused to travel in a
circular path because of the force of friction,
then,...
FN
Fc = FF
FF
car
Fw
Cornering on the Horizontal
Fc = FF
Therefore, mv2/r = mFN
because FN = Fw = mg,
… or,
mv2/r = mmg
FN
FF
car
Fw
m = v2 /rg
Cornering on a Banked Curve
Fp
Fw
FN
Cornering on a Banked Curve
Fc
Fw
Fc
FN
Fw
Q
FN
Note how FN is
the Resultant
Fc
Fw
FN
Q
If we want to know the angle
the curve has to be at to allow
the car to circle without friction,
then we have to analyze the
forces acting on the car.
SinQ = Fc/FN and,
Fc = SinQFN