general physics i 09
DESCRIPTION
General Physics i 09TRANSCRIPT
Chapter 9Center of Mass and Linear
Momentum
Things to learn
We will find simple physical rules for a system of particles by introducing the center of mass.We will reformulate Newton’s 2nd law for a system of particles by introducing linear momentum.We will learn the law of conservation of linear momentum.
We will study collision (inelastic or elastic).We will learn the relation between impulse and linear momentum: impulse-linear momentum theorem.
9-2 The Center of MassYou cannot represent the thrown bat as a tossed point-like object (particle, a ball, etc).
Most objects have non-negligible volume and are a system of particles.
One special point, however, moves in a simple parabolic path.
The special point moves as though (1) the bat’s total mass were concentrated there and (2) the gravitational force on the bat acted only there.
This special point is called the center of mass.
9-2 The Center of Mass
The center of mass of a system of particles (i=1, …,
N) at ri with mass mi is the average position vector weighted by
particle mass.
A system of point particles
1-D example
Translation effect
Solid bodies
SP 9-1
Three particles of masses m1 = 1.2 kg, m2 = 2.5 kg, and m3 = 3.4 kg form an equilateral triangle of edge length a = 140 cm. Where is the center of mass?
cm 83kg 1.7
cm) kg)(70 (3.4cm) kg)(140 5.2()0)(kg 2.1(1 3
1=
++== ∑
=iiicom xm
Mx
cm 58kg 1.7
cm) kg)(120 (3.4kg)(0) 5.2()0)(kg 2.1(1 3
1=
++== ∑
=iiicom ym
My
Center of mass of two systemsof particles
If we know the mass and the center of mass of
each system of particles, we can derive
the center of mass of the combined system.
SP 9-2 superposition
9-3 Newton’s second law for a system of particles
SP 9-3
9-4 Linear Momentum p
9-5 Linear Momentum P of a system of particles
9-7 Conservation of Linear Momentum
If there is no net force on a system of particles,
the linear momentum of the system is conserved.
It is an immediate consequence of
SP 9-5 Explosion (splitting into 2 pieces)
SP 9-6 Ejection ( splitting into 2 pieces )
SP 9-7 Firecracker (splitting into 3
pieces)
SP 9-7 Firecracker (splitting into 3
pieces)
9-6 Collision and Impulse
Impulse-linear momentum theorem
A collision is an isolated event in which two or more bodies exert relatively strong forces on each other for a relatively short time.
CP 5
9-8 Momentum and Kinetic Energy in Collision
Kinetic energy may decrease.
Elastic : If there is NO loss in total kinetic E
Inelastic : If there is any loss in total kinetic E
Total linear momentum is always conserved!
Why? There is NO net external force
Closed system: no mass enters or leaves itIsolated system: no net external forces on it
9-9 Inelastic Collision in 1 Dimension
Momentum Conservation
Completely Inelastic Collision
Momentum Conservation
Stick together(The 2nd particle does not have to
be at rest initially)
Energy loss in inelastic collision
Velocity of Center of Mass
Momentum Conservation
SP 9-8 Ballistic pendulumStep 1
Step 2: Mechanical E conservation
9-10 Elastic Collisions in 1 Dimension
Both momentum and Kinetic energy are conserved in an elastic collision
Elastic CollisionEnergy & Momentum:
conservedCase 1: stationary target
v2i=0
Elastic Collision [Energy & Momentum: conserved]Case 1: stationary target v2i=0
1. EQUAL Mass
A pool player’s resultElastic head-on collision the target will be
at rest after the collision.Kinetic energy is conserved.
Elastic Collision [Energy & Momentum: conserved]Case 1: Stationary target
2. Massive Target
It is like pitching a ball on to the wall.If the target is very massive, it won’t move fast after the collision. Elastic collision conserves the total kinetic energy. So the ball has the same speed after the collision with the direction of its motion flipped.
Elastic Collision [Energy & Momentum: conserved]Case 1: Stationary target
3. Massive Projectile
It is like a pin hit by a bowling ball.
Elastic CollisionEnergy & Momentum:
conservedCase 2: moving target
Elastic CollisionEnergy & Momentum:
conservedCase 2: moving target
9-11 Collisions in 2 Dimensions
9-12 Systems with Varying Mass: A Rocket
M : mass of the rocket, v : its velocity
dM : (negative) change in M, dv : change in v
U : velocity of the exhaust product (-dM)
The system is closed and isolated: the linear momentum must be conserved.
(v of rocket r.t. frame)
= (v of rocket r.t. products) + (v of products r.t. frame)
))(( dvvdMMUdMMv +++−=
equation)rocket (first /
)(
MaRvRdtdMMdvvdMvdvvU
Uvdvv
rel
rel
rel
rel
=−≡=−−+=
+=+
equation)rocket (second lnf
irelif
M
Mrel
v
v
rel
MMvvv
MdMvdv
MdMvdv
f
i
f
i
=−
−=
−=
∫∫