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

=−

−=

−=

∫∫