chapter 9a - systems of particles center of mass –point masses –solid objects newton’s second...
TRANSCRIPT
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