Fundamental Equations of Dynamics
KINEMATICS
Particle Rectilinear Motion
Variable a
Constant a = ac
dv
v = v0 + act
a=
dt
ds
s = s0 + v0t + 12 act 2
v =
dt
a ds = v dv
v2 = v20 + 2ac(s - s0)
Equations of Motion
Particle
"F = ma
Rigid Body
"Fx = m(aG)x
(Plane Motion) "Fy = m(aG)y
"MG = IGa or "MP = "(mk)P
Principle of Work and Energy
T 1 + "U1 - 2 = T 2
Kinetic Energy
Particle
T = 21mv2
Rigid Body (Plane Motion) T = 1mvG2 + 1IG v2
Particle Curvilinear Motion
x, y, z Coordinates
r, u, z Coordinates
#
##
#
#
##
ax = x
vx = x
Work
vr = r ar = r - r u 2
#
##
#
##
# Variable force
#
vy = y
ay = y
vu = r u au = r u + 2r u
#
##
vz = z
az = z
n, t, b Coordinates
#
v = s
Relative Motion
vB = vA + vB/A
#
vz = z
##
az = z
dv
#
at = v = v
ds
[1 + (dy >dx)2]3>2
v2
an =
r =
r
0 d2y >dx 2 0
aB = aA + aB/A
Rigid Body Motion About a Fixed Axis
Variable a
Constant a = ac
dv
a =
v = v0 + act
dt
du
v =
u = u0 + v0t + 21act 2
dt
v dv = a du
v2 = v20 + 2ac(u - u0)
For Point P
s = ur v = vr at = ar an = v2r
Relative General Plane Motion—Translating Axes
vB = vA + vB>A (pin) aB = aA + aB>A (pin)
Relative General Plane Motion—Trans. and Rot. Axis
vB = vA + ! * rB>A + (vB>A )xyz
#
aB = aA + ! * rB>A + ! * (! * rB>A ) +
2! * (vB>A )xyz + (aB>A )xyz
KINETICS
r2 dm
L
I = IG + md2
I
k =
Am
Mass Moment of Inertia I =
Parallel-Axis Theorem
Radius of Gyration
2
UF =
2
F cos u ds
L
(Fc cos u) # s
- W #y
- 1 21 ks 22 - 21 ks 21 2
M#u
Constant force
UF =
Weight
UW =
Spring
Us =
Couple moment
UM =
Power and Efficiency
Pout
Uout
dU
e =
=
P=
= F#v
dt
Pin
Uin
Conservation of Energy Theorem
T1 + V 1 = T2 + V 2
Potential Energy
V = V g + V e, where V g = {W y, V e = + 21 ks 2
Principle of Linear Impulse and Momentum
Particle
mv1 + "
L
F dt = mv2
F dt = m(vG)2
L
Conservation of Linear Momentum
"(syst. mv)1 = "(syst. mv)2
(vB)2 - (vA)2
Coefficient of Restitution
e =
(vA)1 - (vB)1
Principle of Angular Impulse and Momentum
Rigid Body
Particle
m(vG)1 + "
MO dt = (HO)2
L
where HO = (d)(mv)
(HO)1 + "
MG dt = (HG)2
L
where HG = IGv
(HG)1 + "
Rigid Body
(Plane motion)
MO dt = (HO)2
L
where HO = IOv
Conservation of Angular Momentum
"(syst. H)1 = "(syst. H)2
(HO)1 + "