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Physics 218: Mechanics Instructor: Dr. Tatiana Erukhimova
Lectures 25, 26
Second exam Tuesday, October 25
107 Heldenfels
Sections 818 and 819 start at 7 pm
Sections 820 and 821 start at 8:05 pm
Work Energy Theorem
22
222
1
initialfinalr
r
total
mVmVrdFW
22
2222
11
22
11
initialfinalyx
yx
totaly
yx
yx
totalx
mVmVdyFdxF
If a force can be written as the gradient (slope) of some scalar function, that force is conservative.
U(x) is called the potential energy function for the force F
If such a function exists, then the force is conservative
1D case:dx
dUFx
)]()([ 12
2
1
2
1
xUxUdxdx
dU
FdxW
x
x
x
x
veconservati
veconservatiW does NOT depend on path!
1D case:dx
dUF
conW does NOT depend on path!
)]()([ 12 xUxUW con
If Fx(x) is known, you can find the potential energy function as
CdxxFxU x )()(
or
112221 ,0 UKUKW nc If
112221 UKUKW nc
Newton’s 2nd Law
amF
xx maF yy maF
Work does not depend on the choice of coordinates! You can calculate work of each force in the most convenient system of coordinates.
Do not forget to work with components!
Usually there are several ways to solve the problem. Indicate the law or the theorem you use!
Average 76.9Median 81
Sec 818 73.3Sec 819 75.0Sec 820 78.3Sec 821 80.9
Center of Mass
i
N
iicm rm
Mr
1
1
N
iimM
1
i
N
iicm xm
Mx
1
1i
N
iicm ym
My
1
1
Find the position of the center of mass of the system of the sun and Jupiter. (Since Jupiter is more massive than the rest of the planets put together, this is essentially the position of the center of mass of the solar system.) Does the center of mass lie inside or outside the sun?
Motion of the Center of Mass
N
iiicm vm
Mv
1
1
N
iiixcm xvm
Mv
1
1
N
iiiycm yvm
Mv
1
1
i
N
iicm am
Ma
1
1
N
iicm FaM
1
N
iicm FaM
1
The center of mass of a system moves as if all of the mass of the system were concentrated at that point and as if all of the forces were acting at that point
There is only the external forces that affect the motion of the center of mass ( )
external
N
iexternalicm FFaM
1
2112 FF
Only external forces affect the motion of the center of mass
Momentum is a vector!
vmp
cmi
N
ii
N
iitotal vMvmpP
11
N
iiicm vm
Mv
1
1
cmtotal vMP
Vector equation!
dt
Pd
dt
vdMFaM totalcm
externalcm
externaltotal Fdt
Pd
If ,0externalF
0dt
Pd total
ConstPtotal
Conservation of Momentum
If there is no external force on a system, then the total momentum of the system is a constant
ConstPtotal
)()( afterPbeforeP
True in X and Y directions separately!
totalx
totalx Fdt
dP total
y
totaly Fdt
dP