Download Outline Mechanical Systems Kinematics Example Projectile Motion

Outline
CGS 3032
Interactive Modeling and
Animation I
• Review
Feburary 9, 2007
• Today
– Translation
– Rotation
– Scaling
• Project submission guideline
– Name your project with following format:
<lastname>_<firstname>_<project name>
– One zipped file for the project folder
– Mechanical Systems
– Kinematics
• Projectile Motion
– Linear Kinetics
Mechanical Systems
• What is the mechanical system
– Certain portion (a part or the whole) of the
object of interest
• Different types of mechanical systems
– Particle
– A system of particles
– A rigid body
– A system of linked rigid bodies, or an
articulated object
Kinematics
• Location (r)
– Position of a particle at a given time instance
• Velocity (v)
– The rate of change in position or the rate of
displacement
dr
v=
dt
• Acceleration (a)
– The rate of change in velocity
Example
• Calculate new speed
v(t + ∆t ) = v(t ) + a∆t
• Calculate new location
1
r (t + ∆t ) = r (t ) + v (t )∆t + a∆t 2
2
a=
dv
dt
Projectile Motion
• Two dimensional motion
– Only force: the force due to gravity.
– The magnitude of the acceleration: g = 9.8 m/s2
• ay = g and ax = 0
– Do not consider the rotation of the earth
• Suppose initial location (x0, y0), velocity (vx0, vy0)
–
–
–
–
vx(t) = vx0
x(t) = x0 + vx0t
vy(t) = vy0 + gt
y(t) = y0 + vy0t + gt2/2
• Example
1
Linear Kinetics
• Force (f)
– Equals to the product of its mass and
acceleration
f = ma
– Two kinds of forces
• Contact Forces
• Field forces
Linear Kinetics – cont
• Momentum (p)
– The amount of motion the particle possesses
– Equals to the product of its mass and velocity
p = mv
• Kinetic energy
– Energy due to the motion of the particle
T=
Momentum Conservation
• Momentum is conserved for any isolated
collision
• For a collision occurring between two objects
– the total momentum of the two objects before the
collision is equal to the total momentum of the two
objects after the collision
– The momentum lost by object 1 is equal to the
momentum gained by object 2
Types of Collision
• Inelastic collision
– Example - rubber ball on a hard surface
– Conserve momentum but not energy
• Perfectly inelastic collision
– Example - two lumps of clay
– Conserve momentum and the two objects stick
together
• Elastic collision
– Example - two billiard balls with no deformations
– Conserve energy and momentum
• Example
1 2 1
mv = vp
2
2
Collision
• Action of objects striking or coming
together
• Involve forces - change in velocity
• Conserve momentum, and maybe energy
• Example
The Three Laws of Motion
• Law of inertia
– No velocity change if
no external force
• Law of acceleration
f = ma
• Law of reciprocal
actions
– All forces occur in
pairs, which are equal
in magnitude and
opposite in direction
2
Collision of Billiards
• If assuming no spin
– Conserve energy
– Conserve linear momentum
• Procedure
Reading
• http://www.glenbrook.k12.il.us/gbssci/Phys
/Class/BBoard.html
• http://www.lightandmatter.com/html_books
/2cl/ch01/ch01.html
• http://theory.uwinnipeg.ca/physics/index.ht
ml
• Details
3