Chapter 9 Center of Mass and Linear Momentum

In this chapter we will introduce the following new concepts:

-Center of mass (com) for a system of particles -The velocity and acceleration of the center of mass -Linear momentum for a single particle and a system of particles

We will derive the equation of motion for the center of mass, and discuss the principle of conservation of linear momentum

Finally we will use the conservation of linear momentum to study collisions in one and two dimensions and derive the equation of motion for rockets

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The center of mass has been defined using the quations given above so that it has the following property: The center of mass of a system of particles moves as thoughall the system's mass were concentrated there, and that the vector sum of all the external forces were applied there The above statement will be proved later. An example isgiven in the figure. A baseball bat is flipped into the airand moves under the influence of the gravitation force. The center of mass is indicated by the black dot. It follows a parabolic path as discussed in Chapter 4 (projectile motion)All the other points of the bat follow more complicated paths

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. C C

O

m1

m3 m2

F1

F2

F2

x y

z

Newton's Second Law for a System of ParticlesConsider a system of n particles of masses m1,m2 ,m3,...,mn

and position vectors r1,r2 , r3,...,

rn , respectively. The position vector of the center of mass is given by:

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(Like in the case of and exploding rocket)

O

m1

m3 m2

F1

F2

F2

x y

z

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m v

p

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O

m1

m3 m2

p1

p2

p3

x y

z

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Fave

O

m1

m3 m2

p1

p2

p3

x y

z

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If the system is isolated i.e. the net force Fnet = 0 linear momentum is conserved

The conervation of linear momentum is true regardless of the the collision typeThis is a powerful rule thet allows us to determine the results of a collision without knowing the details. Collisions are divided into two broad classes: elastic and inelastic. A collision is elastic if there is no loss of kinetic energy i.e. Ki = K f

A collision is inelastic if kinetic energy is lost during the collision due to conversioninto other forms of energy. In this case we have: K f < Ki

A special case of inelastic collisions are known as completely inelastic. In these collisions the two colliding objects stick together and they move as a single body. In these collisions the loss of kinetic energy is maximum (9-14)

One Dimensional Inelastic CollisionsIn these collisions the linear momentium of the colliding objects is conserved → p1i +

p2i =p1 f +

p2 f

m1

v1i + m1

v2i = m1

v1 f + m1

v2 f

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m

m

m

m

v1i

v1f = 0 v2f

v2i = 0 x

x

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v2i = 0

m1

m1

m2

m2

v1i

v1f

v2f

x

x

m1

m1 m2

m2

v1i

v1f v2f

v2i = 0 x

x

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vf

vi Mi/Mf

O (9-22)