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Chapter 9, System of Particles

- Center of Mass
- Linear Momentum and Conservation
- Impulse
- Rocket

Center of Mass for a System of Particles

The center of mass of a body or a system of

bodies moves as though all of the mass were

concentrated there and all external forces were

applied there.

Where is the center of mass of this meter

stick? (a) 30cm (b) 50cm (c) 90cm

The center of mass of a symmetric, homogeneous

body must lie on its geometric center.

2 bodies, 1 dimension

For example, if x10, x2d, and m22m1, we find

that

Example 1

Center of Mass for a System of Particles

2 particles, 1 dimension

n particles, 3 dimensions

n particles, 3 dimensions, vector equation

Example 2

Center of Mass for a Solid Body

dm is the differential mass element

Uniform density

Newtons 2nd Law for a System of Particles

The center of mass moves like an imaginary

particle of mass M under the influence of the net

external force on the system.

A firework rocket explodes

Linear Momentum and Impulse

System

ImpulseLinear Momentum Theorem Impulse Force

? Duration of the force Change in Momentum

Collision of two particle-like bodies

Conservation of Linear Momentum

If no net external force acts on a system of

particles, the total linear momentum P of the

system cannot change.

If the component of the net external force on a

closed system is zero along an axis, then the

component of the linear momentum along that axis

cannot change.

- Closed system (no mass enters or leaves)
- Isolated system (no external net force)

Before collision After collision Sum of

initial momentums Sum of final momentums

- Closed system (no mass enters or leaves)
- Isolated system (no external net force)

Collisions in closed, isolated systems

- Elastic
- Inelastic
- Completely inelastic

- both colliding objects does not change their

shapes - total linear momentum is conserved
- total kinetic energy is conserved

- the shape of one or both of object changes

during the collision - total linear momentum is conserved
- total kinetic energy is NOT conserved

- the colliding objects stick together after the

collision - total linear momentum is conserved
- total kinetic energy is NOT conserved

In a closed, isolated system containing a

collision, the linear momentum of each colliding

body may change but the total momentum P of the

system cannot change, whether the collision is

elastic or inelastic.

Completely Inelastic Collisions in 1D

Velocity of Center of Mass

Sample Problem 9-8

gt

mM

Elastic Collisions in 1D

Equ.s (9-67), (9-68)

Sample Problem 9-10

Rocket Propulsion

A Rocket System (rocket its ejected combustion

products) is a closed, isolated system. The

rocket is accelerated as a result of the thrust

it receives from the ejected gases.

Rocket Equations

First rocket equation

where R is the fuel consumption rate, vrel is

the velocity of ejected fuel with respect to the

rocket M is the instantaneous mass of the

rocket a is the acceleration of the rocket T is

referred to as the thrust of the rocket engine

Second rocket equation

where vi and vf are initial and final velocities

of the rocket Mi and Mf are the initial and

final masses of the rocket

Summary 1

Summary 2

Conservation of momentum

(closed, isolated system)