Particle Kinematics: Intro to Curvilinear Motion

Particle Kinematics: Intro to Curvilinear Motion

Current unit: Particle Kinematics
Last two classes: Straight Line Motion

Today: 3
(a) Intro to Curvilinear Motion
(b) Circular Motion an
(c) Projectile Motion
v3
v1 at a3
a1 v2

an at

1 at = a2
= Radius of an = 0 at an
Cur vature
inflection point
=

Introduction to v1 v2
Curvilinear
Motion v3
If know y = f(x) path,
Key Principle #1:
Velocity is can use it to get v direction (angle):
always
tangent to v
the path. v= v

= tan-1 y

Know y = f(x)

Knowing this… If know vx and vy (as in projectile motion),
Can use path slope can use these to get v and the angle, .

to get velocity angle. vy v v = vx2 + v2y
or,

Can use velocity vx vy
components to get vx
path slope. = tan-1

Introduction to v1 at v2 a3 an
Curvilinear a1 at
Motion
an at = a2 v3
Key Principle #2:
Acceleration at inflection point
always acts
toward the an = 0
concave side
of the curve. The accel vectors a1 and a3 act toward
the concave side of the curve.

If we resolve the a’s into an and at components:
at components act tangent to the curve
an components act to v,
toward the concave side of the curve.

Significance of the at and an components?
at components change the length (speed) of v.
an components change the direction of v.

Intro to Key Principle #3: an acts toward the center of
Curvilinear curvature and may be calc’d from an = v2/ρ
Motion

v1 at 3
a1
v2 a3 an
an at

1 at = a2 v3
= Radius of
Cur vature an = 0 at an

inflection point
=

= 1 + (y )2 3/2 Normal an = v2
y Accel, an

= “radius of curvature” an acts to v,
of a general curve, y = f(x) toward center of curvature.

where, y = f (x) and y = f (x) Changes v direction.