Transcript Two-Dimensional Motion

Two-Dimensional Motion
Projectile Motion
Periodic Motion
Projectile Moion
Vx
Vx
Vy
Vx
Vy
Vx = constant
Vy = varying
Vy
Vx
Vx
Vy
Vx
Vy
Formulas:
Vx = constant
therefore,
Vx = d/t
Vy = varying
therefore, acceleration
vf = vi + at
vf2 = vi2 + 2ad
d = vi + 1/2at2
Vy
Projectile Motion
vi
q
vx
vy
Vy = sinq(vi)
Vx = cosq(vi)
Vy controls how long
it’s in the air and how
high it goes
Vx controls how far it goes
Projectile Motion
“Range formula”
Remember!!!!!
vi
yi
vi is the velocity at an angle and the
sin2q is the sine of 2 x q
R = vi2 sin2q/g
Range formula works only when yi = yf
yf
Projectile Motion
“Range formula”
R = vi2 sin2q/g
vi
yi
If vi = 34 m/s and q is 41o then,
R = (34
m/s)2
sin82o/9.8
m/s2
R = 1160 m2/s2 (0.99)/9.8 m/s2
R = 120 m
yf
Projectile Motion
“Range formula”
Note that if q becomes the complement
of 41o, that is, q is now 49o, then,
vi
q
vi = 34 m/s and q is 49o then,
R = (34 m/s)2 sin98o/9.8 m/s2
R = 1160 m2/s2 (0.99)/9.8 m/s2
R = 120 m
So, both 41o and 49o yield “R”
Projectile Motion
“Range formula”
OR,
vi
yi
q vy
If vi = 34 m/s and q is 41o then,
vx
vy = sin41o(34m/s) = 22m/s, and
t = vfy - viy/g = -22m/s - (22m/s)/-9.8m/s2 = 4.5 s
vx = cos41o(34 m/s) = 26 m/s, and
dx = vx(t) = 26m/s (4.5 s) = 120 m
yf
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
q
v2
r
q
r
v2
Dv = v2 - v1
or Dv = v2 + (-v1)
(-v1) = the opposite of v1
v1
(-v1)
v1
v1
q
v2
r
r
0
v2
Dv = v2 - v1
or Dv = v2 + (-v1)
(-v1) = the opposite of v1
v1
Dv
v2
q
(-v1)
(-v1)
Note how Dv is directed
toward the center of the
circle
v1
Dl
r
q
v1
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
q
Dl
r
q
v2
v2
r
v2
Dv
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
Simple Harmonic Motion
or
S.H.M.
Simple Harmonic motion is motion that has force and
acceleration always directed toward the equilibrium position
and has its maximum values when displacement is maximum.
Velocity is maximum at the equilibrium position and zero at
maximum displacement
Pendulum motion, oscillating springs (objects), and elastic
objects are examples
Simple Harmonic Motion
F = max
a = max
v=0
F = less
a = less
v = greater
F=0
a=0
v = max
F = greater
a = greater
v = less
F = max
a = max
v=0
Force
acceleration
Pendulum Motion
Note that FT (the accelerating force
is a component of the weight of the
bob that is parallel to motion (tangent
to the path at that point).
Fw
FT
Pendulum Motion
Fw
FT
Note that as the arc becomes less
so does the FT, therefore the force
and resulting acceleration also
becomes less as the “bob”
approaches the equilibrium
position.
Pendulum Motion
ac = 4p2 r/T2
ac = g and r = l
g = 4p2 l/T2
T2 = 4p2 l/g
Fw
T = 2p
FT
l/g
Oscillating Elastic Objects
Fe = max
Fe = less
Fe = less
Fe = max
a = max
a = less
a = less
a = max
a and F
=0
Note that no part of Fw is in the
direction on Motion, or FT
There, F and a is zero!!!
FT
Fw