fisica generale - alan giambattista, betty mccarty richardson copyright © 2008 – the mcgraw-hill...
TRANSCRIPT
1
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Chapter 7: Linear Momentum
•Definition of Momentum
•Impulse
•Conservation of Momentum
•Center of Mass
•Motion of the Center of Mass
•Collisions (1d, 2d; elastic, inelastic)
2
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§7.1 Momentum
Consider two interacting bodies:
F21 F12
If we know the net force on each body then tm
t netFav
The velocity change for each mass will be different if the masses are different.
3
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
tm 2111 Fv
Rewrite the previous result for each body as:
ttm 211222 FFv
Combine the two results:
ifif mm
mm
222111
2211
vvvv
vv
4
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The quantity (mv) is called momentum (p).
p = mv and is a vector.
The unit of momentum is kg m/s; there is no derived unit for momentum.
5
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
From slide (3):
21
2211
pp
vv
mm
The change in momentum of the two bodies is “equal and opposite”. Total momentum is conserved during the interaction; the momentum lost by one body is gained by the other.
6
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§7.2 Impulse
Definition of impulse: t Fp
One can also define an average impulse when the force is variable.
7
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text conceptual question 7.2): A force of 30 N is applied for 5 sec to each of two bodies of different masses.
30 N
m1 or m2
Take m1<m2
(a) Which mass has the greater momentum change?
t FpSince the same force is applied to each mass for the same interval, p is the same for both masses.
8
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(b) Which mass has the greatest velocity change?
Example continued:
m
pv
Since both masses have the same p, the smaller mass (mass 1) will have the larger change in velocity.
(c) Which mass has the greatest acceleration?
t
v
aSince av the mass with the greater velocity change will have the greatest acceleration.
9
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.9): An object of mass 3.0 kg is allowed to fall from rest under the force of gravity for 3.4 seconds. What is the change in momentum? Ignore air resistance.
Want p =mv.
(downward) m/s kg 100p
m/sec 3.33
vm
tgv
tav
10
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.10): What average force is necessary to bring a 50.0-kg sled from rest to 3.0 m/s in a period of 20.0 seconds? Assume frictionless ice.
N 5.7
s 0.20
m/s 0.3kg 0.50
av
av
av
F
t
m
t
t
vpF
Fp
The force will be in the direction of motion.
11
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§7.3 Conservation of Momentum
v1i
v2i
m1 m2
m1>m2
m1 m2
A short time later the masses collide.
What happens?
12
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
During the interaction:
N1
w1
F21
N2
w2
F12
x
y
1121
11 0
amFF
wNF
x
y
2212
22 0
amFF
wNF
x
y
There is no net external force on either mass.
13
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The forces F12 and F21 are internal forces. This means that:
ffii
ifif
2121
2211
21
pppp
pppp
pp
In other words, pi = pf. That is, momentum is conserved. This statement is valid during the interaction only.
14
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.18): A rifle has a mass of 4.5 kg and it fires a bullet of 10.0 grams at a muzzle speed of 820 m/s. What is the recoil speed of the rifle as the bullet leaves the barrel?
As long as the rifle is horizontal, there will be no net external force acting on the rifle-bullet system and momentum will be conserved.
m/s 1.82m/s 820kg 4.5
kg 01.0
0
br
br
rrbb
fi
vm
mv
vmvm
pp
15
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§7.4 Center of Mass
The center of mass (CM) is the point representing the mean (average) position of the matter in a body. This point need not be located within the body.
16
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
The center of mass (of a two body system) is found from:
21
2211
mm
xmxmxcm
This is a “weighted” average of the positions of the particles that compose a body. (A larger mass is more important.)
17
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
In 3-dimensions write
ii
iii
cm m
m rr where
iimmtotal
The components of rcm are:
ii
iii
cm m
xmx
ii
iii
cm m
ymy
ii
iii
cm m
zmz
18
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.27): Particle A is at the origin and has a mass of 30.0 grams. Particle B has a mass of10.0 grams. Where must particle B be located so that the center of mass (marked with a red x) is located at the point (2.0 cm, 5.0 cm)?
x
y
A
x
ba
bb
ba
bbaacm mm
xm
mm
xmxmx
ba
bb
ba
bbaacm mm
ym
mm
ymymy
19
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
cm 8
cm 2g30g 10
g 10
b
bcm
x
xx
cm 20
cm 5g 10g 30
g 10
b
bcm
y
yy
Example continued:
20
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.30): The positions of three particles are (4.0 m, 0.0 m), (2.0 m, 4.0 m), and (-1.0 m, -2.0 m). The masses are 4.0 kg, 6.0 kg, and 3.0 kg respectively. What is the location of the center of mass?
x
y
3
2
1
21
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
m 92.1
kg 364
m 1kg 3m 2kg 6m 4kg 4321
332211cm
mmm
xmxmxmx
m 38.1
kg 364
m 2kg 3m 4kg 6m 0kg 4321
332211cm
mmm
ymymymy
Example continued:
22
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§7.5 Motion of the Center of Mass
For an extended body, it can be shown that p = mvcm.
From this it follows that Fext = macm.
23
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.35): Body A has a mass of 3.0 kg and vx = +14.0 m/s. Body B has a mass of 4.0 kg and has vy = -7.0 m/s. What is the velocity of the center of mass of the two bodies?
Consider a body made up of many different masses each with a mass mi.
The position of each mass is ri and the displacement of each mass is ri = vit.
24
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
For the center of mass:
ii
iii
m
mt
rvr cmcm
Solving for the velocity of the center of mass:
ii
iii
ii
i
ii
m
m
mt
m vr
vcm
Or in component form:
ii
ixii
m
vm ,
xcm,v
ii
iyii
m
vm ,
ycm,v
Example continued:
25
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Applying the previous formulas to the example,
m/s 6
kg 43
m/s 0kg 4m/s 14kg 3
,,,
ba
xbbxaaxcm mm
vmvmv
m/s 4
kg 43
m/s 7kg 4m/s 0kg 3
,,,
ba
ybbyaaycm mm
vmvmv
Example continued:
26
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
§§ 7.6 & 7.7 Collisions in One and Two Dimension
When there are no external forces present, the momentum of a system will remain unchanged. (pi = pf)
If the kinetic energy before and after an interaction is the same, the “collision” is said to be perfectly elastic. If the kinetic energy changes, the collision is inelastic.
27
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
If, after a collision, the bodies remain stuck together, the loss of kinetic energy is a maximum. This type of collision is called perfectly inelastic.
28
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.41): In a railroad freight yard, an empty freight car of mass m rolls along a straight level track at 1.0 m/s and collides with an initially stationary, fully loaded, boxcar of mass 4.0m. The two cars couple together upon collision.
(a) What is the speed of the two cars after the collision?
m/s 2.0
0
121
1
212111
2121
vmm
mv
vmmvmvmvm
pppp
pp
ffii
fi
29
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
(b) Suppose instead that both cars are at rest after the collision. With what speed was the loaded boxcar moving before the collision if the empty one had v1i = 1.0 m/s.
m/s 25.0
00
12
12
2211
2121
ii
ii
ffii
fi
vm
mv
vmvm
pppp
pp
Example continued:
30
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.49): A projectile of 1.0 kg mass approaches a stationary body of 5.0 kg mass at 10.0 m/s and, after colliding, rebounds in the reverse direction along the same line with a speed of 5.0 m/s. What is the speed of the 5.0 kg mass after the collision?
m/s0.3m/s 5.0m/s 10kg 0.5
kg 0.1
0
112
12
221111
2121
fif
ffi
ffii
fi
vvm
mv
vmvmvm
pppp
pp
31
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Example (text problem 7.58): Body A of mass M has an original velocity of 6.0 m/s in the +x-direction toward a stationary body (B) of the same mass. After the collision, body A has vx=+1.0 m/s and vy=+2.0 m/s. What is the magnitude of body B’s velocity after the collision?
A
B
vAi
Initial
B
AFinal
32
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
fxfxix
fxfxixix
fxix
vmvmvm
pppp
pp
221111
2121
0
fyfy
fyfyiyiy
fyiy
vmvm
pppp
pp
2211
2121
00
x momentum:
Solve for v2fx: Solve for v2fy:
m/s 00.5
11
2
11112
fxix
fxixfx
vv
m
vmvmv
y momentum:
m/s 00.2
12
112
fyfy
fy vm
vmv
m/s 40.522
22
2 fxfyf vvvThe mag. of v2 is
Example continued:
33
Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Copyright © 2008 – The McGraw-Hill Companies s.r.l.
Summary
•Definition of Momentum
•Impulse
•Center of Mass
•Conservation of Momentum