Chapter 3 Momentum

i
i p mvi Mvc
p pi mvi
i
The linear momentum of a system of particles is equal to the product of the total mass M of the system and the velocity of the center of mass.
F1
F21 F12
m1
F2
m2
Imagining a collection of objects that are moving through space and are not toughing one another. For the ith object
ex in d (mi v i) Fi Fi dt
Consider the composite disk at right, with a main section of mass density ρ and radius d, and a subsection of mass density 2ρ and radius d/2. If O marks the origin of our coordinate system, at the center of the composite disk, where is the disk’s center of mass located?
3.3 Conservation of linear momentum
System of objects(系统）：n interacting objects Internal force (内力)：Objects within the system exert on other objects in the system. External force(外力)：Forces on objects of the system are exerted by objects outside the system
c c c
c c c c
General motion =translational motion of com+ rotation about the com 其余点的运动=随C点的平动+绕C点的转动
2. Com of a system of particles
m 1 r1 m 2 r2 m i ri rC m1 m 2 m i
2. Impulse
Unit：N·s Differential form
dp d(mv) Fnet dt dt Fnet dt dp d (mv)

t2
t1
Fnet dt p2 p1 mv2 mv1
Impulse:
t2 J Fdt
Fin total 0

t2
t1
p2 Fextotal dt dP p2 p1
J total p2 p1
p1
Impulse acted by net force=change in momentum of system
if Fext 0, P = Pi =constant
O
3 rC hk 4
Determine com of half uniform spherical shell with radius R.
Rsin θ
y
Rdθ
R
θ dθ O
Rcos θ
x
Solution: Choose the coordinate system as shown Area of the circular belt From symmetry，so xc= 0
The sum of all the forces acting on the system of particles ( or extended body) is equal to the total mass of the system times acceleration of its center of mass. 作用在系统上的合外力等于系统的总质量乘以质心 的加速度——质心运动定律 That is the system moves as if all its mass were concentrated at the com and all the external forces acted at that point.
(1) J has the same direction with that of p p2 p1
⑵component form
质点所受合力的冲量=质点动量的增量 ----质点的动量定理
J x p2 x p1x J y p2 y p1 y J p p 2z 1z z
Sum of all terms all the internal forces cancel each other in pairs
Fext is the sum of all external forces acting on our system.
dptotal d d Fextotal (mi vi ) (mi vi ) dt i dt i dt
Differentiate this equation with respect to time n n drc dri M m i M vC mi vi dt i 1 dt i 1

Take the derivative with respect to time again n n dvc M m i ai F i dt i 1 i 1 dvc Fext Mac M dt
y
o
r
mi dm
z
x
1 rc rdm M
For symmetrically shaped bodies of uniform composition such as spheres, cylinders, and rectangular solids, the com is located at the geometric center of the body.
yC R cos sin d R 2 com： rC R 2 j
π 2 0
x
4. Newton’s second law for a system of particles
rc m i ri
i 1 n
M
n Mrc m i ri i 1
Key word and Terms
impulse momentum linear momentum impulsive force isolated system center of mass law of conservation of momentum symmetrically 冲量 动量 线动量 冲力 孤立系统 质心 动量守恒定律 对称地
* 3.4 Systems with varying mass: A rocket
The effect of force over a time interval
J,
P
Momentum ， Impulse ， momentum theorem Law of conservation of linear momentum
dA 2 πR sin Rd
dm 2πR sin d
2
How about a half solid sphere with radius R? rC 3R 8 j
Rsin θ
y
Rdθ
R
θ
dθ
Rcos θ
y R cos θ 2 y 2π R sin d 1 yC ydm 2 2πR M
Determine the com of a uniform cone of height h and radius R as shown. Solution: Take the coordinate system as shown
1 zc zdm M
2
From symmetry
z
r
xc 0, yc 0
Chapter 3 Conservation of linear momentum and collisions 3.1 Center of Mass and translational motion
1. Center of mass (com) The motion trajectory of point C is a parabola.
t1
Integral form
Definition: The integral of the force over the time interval
Discussion: (1)Impulse is a vector, also a process quantity. (2)The average force over a time interval
dvc Fext Mac M dt
3.2 Linear momentum and impulse
1. linear momentum A particle
p mvቤተ መጻሕፍቲ ባይዱ
Unit：kg ·m/s
Linear momentum is a instantaneous vector，its direction is the same with that of the velocity. A system of n objects