Kinematics (the geometry of the motion)
Basic kinematic relationships
For a point
dx
v=
= x&
dt
a=
dv d 2 x
= 2 = &&
x
dt
dt
or
For a rigid body
dθ &
ω=
=θ
dt
a=v
dv
dx
(From using
the chain rule)
α=
dω d 2 θ &&
= 2 =θ
dt
dt
or
α=ω
dω (From using
dθ the chain rule)
To obtain algebraic relationships separate variables and integrate. For example:
Constant acceleration equations (useful for any problem with constant acceleration - such as
projectile motion):
a = constant
α = constant
1
1
x = x0 + v0t + at2
θ = θ0 + ω 0t + αt 2
2
2
v = v0 + at
ω = ω 0 + αt
v 2 = v02 + 2a ( x − x0 )
ω 2 = ω 20 + 2α ( θ − θ0 )
We can also express the velocity and acceleration in various coordinate systems. This is helpful
when we apply linear momentum.
Rectangular components
r
v = v x $i + v y $j
r
a = a x $i + a y $j
Normal and tangential components
r
v = ve$ t
v2
r
a = v&$e t +
e$ n
ρ
Radial and transverse components
(
r
&$r + rθ& e$θ
v = re
)
(
)
r
θ + 2r&θ& e$θ
a = r&& − rθ& 2 e$r − r&&
Other topics:
Relative Motion
r
r
r
v A = v B + v A/ B
r
r
r
a A = a B + a A/ B
vB
vA/B
vA
(Vector Equations)
Dependent Motion
1) Define coordinates, 2) write constraint equations, 3) differentiate
Rigid Body Kinematics
Translation - All point on the body have the same velocity and acceleration
Fixed axis rotation (i.e. find the velocity and acceleration of any point on the rigid body)
v = ωrp/o
velocity: magnitude = ωr
P
direction = perpendicular to r
rp/o
O
at = αrp/o
acceleration: tangential component = αr
normal component = ω2r
P
O
an = ω2rp/o
General plane motion (always valid for two points on the same rigid body)
r
r
r
a A = a B + a A/ B
r r
r
r r
r
= a B + α × r A / B + ω × ( ω × rA / B )
r r
r
r
= a B + α × rA / B − ω 2 rA / B (plane motion)
r
r
r
v A = v B + v A/ B
r r
r
= v B + ω × rA / B
To solve the velocity problem it is often easiest to instantaneous centers. Instantaneous center
cannot be used for accelerations!
Instantaneous Center (draw lines perpendicular to velocity)
IC