Center&of&mass& · Whatcan&the&velocity&of¢er&of&mass&tell&us&& aboutmomentum&?& 5 4 3 2 1 0 1...
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Center of mass
Transcript of Center&of&mass& · Whatcan&the&velocity&of¢er&of&mass&tell&us&& aboutmomentum&?& 5 4 3 2 1 0 1...
Center of mass
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
1 Kg
1 Kg
Center of mass (a) -‐3 (b) 0 (c) 6
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
1 Kg
1 Kg
1 Kg
Center of mass (a) 0 (b) 3 (c) 1
Formula for center of mass posi?on
xcm = m1x1+m2x2+···+mkxkm1+m2+···+mk
xcm = m1x1+m2x2m1+m2
xcm = m1x1+m2x2+m3x3m1+m2+m3
Two bodies
Three bodies
General
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
4 Kg
5 Kg
Center of mass is at 0. What is the blue mass ?
??
Ans: 3 Kg
x
y
3 Kg
2 Kg 5 Kg
Find the Center of Mass
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
4 Kg
8 Kg
2 m/s t=0
What is the velocity of the center of mass ?
x1 = −3 + 2t
x2 = 3
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
4 Kg
8 Kg
2 m/s
At ?me t x1 = −3 + 2t
x2 = 3
⇒ vcm = 23m/s
xcm =4(−3 + 2t) + 8(3)
4 + 8
=12 + 8t
12
= 1 +2
3t
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
4 Kg
8 Kg
2 m/s
Is the average velocity of the two blocks ? vcm
Weighted average velocity = 4(2)+8(0)4+8 = 8
12 = 23 m/s
So velocity of the center of mass is the weighted average of the veloci?es of the blocks
What can the velocity of center of mass tell us about momentum ?
-‐4 0 1 2 3 4 5 -‐1 -‐2 -‐3 -‐5
4 Kg
8 Kg
2 m/s
Total momentum = PT = 4(2) + 8(0) = 8 kgm/s
Total mass = MT = 4 + 8 = 12 kg
So we observe that
PT = MT vcm
Momentum in 2-‐d
A 1 Kg
B 2Kg
2 m/s
3 m/s
A 1 Kg
B 2Kg
Find magnitude and direc?on of final velocity
A 1 Kg
B 2Kg
2 m/s
3 m/s
4
3 5
cos θ = 45
velocity = 5/3 m/s
momentum vector
A 2 Kg
B 4 Kg
2 m/s
3 m/s
A 2 Kg
B 4Kg
Find magnitude and direc?on of final velocity
45 o
30 o
A 2 Kg
B 4 Kg
2 m/s
3 m/s
A 2 Kg
B 4Kg
45 o
30 o
+
+
1 Kg
1 Kg
1 Kg
1 Kg
20 m/s
10 m/s
??
45 o
What is the angle at which the red ball scaWers ?
Higgs Boson simula?on
