Post on 17-Jan-2016
Two-Dimensional Motion
Projectile MotionProjectile Motion
Periodic MotionPeriodic Motion
Projectile MoionVx
Vx
Vx
Vy
Vy
Vy
Vx = constant
Vy = varying
Vx
Vx
Vx
Vy
Vy
Vy
Formulas:
Vx = constant therefore,
Vx = d/t
Vy = varying therefore, acceleration
vf = vi + atvf
2 = vi2 + 2ad
d = vi + 1/2at2
Projectile Motion
vi vy
vx
Vy = sin(vi)
Vx = cos(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”
vi R = vi2 sin2/g
yiyfRange formula works only when yi = yf
Remember!!!!! vi is the velocity at an angle and the sin2 is the sine of 2 x
Projectile Motion“Range formula”
vi
R = vi2 sin2/g
yiyf
If vi = 34 m/s and is 41o then,
R = 1160 m2/s2 (0.99)/9.8 m/s2
R = 120 m
R = (34 m/s)2 sin82o/9.8 m/s2
Projectile Motion“Range formula”
vi
Note that if becomes the complementof 41o, that is, is now 49o, then,
vi = 34 m/s and is 49o then,
R = 1160 m2/s2 (0.99)/9.8 m/s2
R = 120 m
R = (34 m/s)2 sin98o/9.8 m/s2
So, both 41o and 49o yield “R”
Projectile Motion“Range formula”
vi
yiyf
OR,
If vi = 34 m/s and is 41o then,vyvy = sin41o(34m/s) = 22m/s, and
t = vfy - viy/g = -22m/s - (22m/s)/-9.8m/s2 = 4.5 s
vx
dx = vx(t) = 26m/s (4.5 s) = 120 m
vx = cos4 m/s) = 26 m/s, and
Circular Motion
When an object travels about a When an object travels about a given point at a set distance it is said given point at a set distance it is said to be in to be in circular motioncircular motion
Cause of Circular Motion
11stst Law…an object in motion stays in motion, in Law…an object in motion stays in motion, in
a straight line, at a constant speed unless acted a straight line, at a constant speed unless acted on by an outside force.on by an outside force.
22ndnd Law…an outside force causes an object to Law…an outside force causes an object to accelerate…a= F/maccelerate…a= F/m
THEREFORETHEREFORE, circular motion is caused by a , circular motion is caused by a force that causes an object to travel contrary to force that causes an object to travel contrary to its inertial pathits inertial path
Circular Motion Analysisv1
v2
rr
Circular Motion Analysis
v1
v2
rr
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
Because the two triangles aresimilar, the angles are equal andthe ratio of the sides areproportional
l
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
Simple Harmonic Motionor S.H.M.
Simple Harmonic motion is motion that has force and acceleration always directed toward the equilibrium positionand has its maximum values when displacement is maximum.Velocity is maximum at the equilibrium position and zero atmaximum displacement
Pendulum motion, oscillating springs (objects), and elastic objects are examples
F = maxa = maxv = 0
F = maxa = maxv = 0
F = greatera = greaterv = lessF = 0
a = 0v = max
F = lessa = lessv = greater
Simple Harmonic Motion
Force
acceleration
Pendulum Motion
Fw
FT
Note that FT (the accelerating forceis a component of the weight of thebob that is parallel to motion (tangentto the path at that point).
Pendulum Motion
Fw
FT
Note that as the arc becomes lessso does the FT, therefore the force and resulting acceleration also becomes less as the “bob” approaches the equilibrium position.
Pendulum Motion
Fw
FT
ac = r/T2
ac = g and r = l
g = l/T2
T2 = l/g
T = 2 l/g
Oscillating Elastic ObjectsFe = max Fe = maxFe = less Fe = less
a = max a = maxa = lessa = less
a and F = 0
Fw
FT
Note that no part of Fw is in thedirection on Motion, or FT
There, F and a is zero!!!