Chapter 9a - Systems of Particles

• Center of Mass– point masses

– solid objects

• Newton’s Second Law for a System of Particles

• Linear Momentum for a System of Particles

• Conservation of Linear Momentum– Rockets

– Internal Energy/External Forces

Calculating the center of mass – point objects – 1 D

n n

i i i i1 1 2 2 i 1 i 1

cm n1 2

ii 1

m x m xm x m x ...

xm m ... Mm

Calculating the center of mass – point objects – 2 D

n

i ii 1

cm

m xx

M

n

i ii 1

cm

m yy

M

n

i ii 1

cm

m zz

M

n

i ii 1

cm

m rr

M

Problem 1

• Three masses located in the x-y plane have the following coordinates:– 2 kg at (3,-2)– 3 kg at (-2,4)– 1 kg at (2,2)

• Find the location of the center of mass

Calculating the center of mass – solid objects – 1 D

n

i ii 1

cm

m xx

M

cm

xdmx

M

Calculating the center of mass – solid objects – 2 D

cm

xdmx

M

cm

ydmy

M

cm

zdmz

M

cm

rdmr

M

Finding the COM

Problem 2

• What is the center of mass of the Letter “F” shown if it has uniform density and thickness?

20cm

15cm

10 cm

2cm

2cm

2cm

5cm

Problem 3

The blue disk has a radius 2R

The white area is a hole in the Disk with radius R.

Where is the center of mass?

COM and translational motionn

i ii 1

cm

m rr

M

n

cm i ii 1

Mr m r

ncm i

ii 1

dr drM m

dt dt

First time derivative

n

cm i ii 1

Mv m v

COM Momentum

Second time derivativen

cm ii

i 1

dv dvM m

dt dt

n n

cm i i 1 2 3 ii 1 i 1

Ma m a F F F ... F

Newton’s 2nd Law

What this means….

• The sum of all forces acting on the system is equal to the total mass of the system times the acceleration of the center of mass.

• The center of mass of a system of particles with total mass M moves like a single particle of mass M acted upon by the same net external force.

Conservation of Linear Momentum

• If 2 (or more) particles of masses m1, m2, … form an isolated system (zero net external force), then total momentum of the system is conserved regardless of the nature of the force between them.

Problem 1

• An astronaut finds himself at rest in space after breaking his lifeline. With only a space tool in his hand, how can he get back to his ship which is only 10 m away and out of his reach.

Variable mass – Rocket propulsion

eM dM v M v dv v dv v dM

eMv vdM Mv Mdv vdM dvdM v dM

fip p

small

eMdv v dM

e

dv dMThrust M v

dt dt

Rocket thrust

rel

dv dMThrust M v

dt dt

0m 21000kg

fuelm 15000kg

dM190kg / s

dt

ev 2800m / s

Find: Thrust, initial net force,net force as all fuel expended