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