slide 1Physics 1401 - L 9 Frank Sciulli

Lecture 9

l Midterm 1 returned today (shelf outside 301 entry. Discussed statistics.

l See website for the exam, the solutions, and the statistics associated with the grading.u Partial credit has been allotted in a consistent manner.u If you believe there has been an error made in grading

your exam, fill in a regrade request, available fromUgrad secretary in the Physics office. (See **exam specifics webpage.) Any request must be made within two weeks of today.

l Today … continue potential energy functions and conservative forces; then go on to chapter 9 on how to apply what we know so far to complex systems.

l Read carefully chapters 9 and 10

slide 2Physics 1401 - L 9 Frank Sciulli

Conservative Force? How do we tell?

l Recall movie “work by grav”l Conservative force: doesn’t depend on path

ab

ab

gr

b

av

spring

a

U U U

W

U y mgy

U x

U dx

kx

F

212

( )

( )

∆ ≡ −

− =

=

=

∆ ≡ −∫If depends on path, the potential cannotbe defined and force is non-conservative!

review

slide 3Physics 1401 - L 9 Frank Sciulli

Work Energy Theorem →Conservation of Mechanical Energy

Pendulum illustrates conservation of mechanical energy

1 1 2

2

2

2

1

1

W K KU U

K U KW

U

∆ = −

− = −

+ = +

∆

212(1 cos ) .mgL mv constθ− + =

Note that doesn’t answer all questions! For x(t), better to use forces!

review

slide 4Physics 1401 - L 9 Frank Sciulli

Pendulum Problem

y

y L LU mgLK mvU K E

θ

θ21

2

cos(1 cos )

is consta t n

= −

= −

=

+ =

review

l Describe the motion of a pendulum

l Can be done by using forces

l Some aspects can be done by using energy

slide 5Physics 1401 - L 9 Frank Sciulli

Changing work into energy in real world

Man gets kinetic energy by doing work pushing against the ground

Kinetic energy transformed into “spring” potential energy by deforming pole

Pole potential “spring” energy becomes man’s gravitational potential energy

review

slide 6Physics 1401 - L 9 Frank Sciulli

8-35: a problem involving conservative force (gravity)

Boy starts at top of semispherical mound of ice (no friction) and slides down. What angle with vertical when he leaves the ice?

θ θ

N

mg

(1 cos )R θ−

048.2θ =

slide 7Physics 1401 - L 9 Frank Sciulli

Force from Potential Energy Function (sec 8-5)

Two Important Uses of Potential Energyu conservation of

mechanical energy

u Potential function to determine force

ab

bxa

ab

x

xx

U U UU F x

U W F dx

U dUFx d

x x xU F x

x∆ →

∆= −

∆ ≡ −

∆ = − = −

∆ =

= −∆

∆ = − ∆

∫

0

For & varying with

so for small changes, - which requires

limSimple and direct way to find the relevant force!

slide 8Physics 1401 - L 9 Frank Sciulli

Graphs P. E. for Gravity and Springs

Relationship between force and potential functions for two cases:gravityspring

xydUFd

dUFdy x

== − −

unbounded bounded

Potential“WELL”

motion

slide 9Physics 1401 - L 9 Frank Sciulli

Real World: Topological Map and Equipotentials

l Gives elevation (h) versus x and y.l But Ugrav~ mgh so these curves are equipotentialsl Net forces can be determined ---l Take s alongterrain … then force normal to terrain has component along terrain equal to …….

sU hF mgs s

∆ ∆= − = −

∆ ∆

slide 10Physics 1401 - L 9 Frank Sciulli

Generalization of Force from Potential Function

So knowledge of potential function versus position tells us about the force!

Has broad implications! Cultural for now!

Suppose Potential Function depends on 3D:

( ,

ˆˆ ˆ

( , , ) called "gr

, )

adient"

U U UF i j kx y z

F U x y

x

z

U y z∂ ∂ ∂

= − − −∂ ∂ ∂

= −∇

∇

r

r rr

Aside from mathwhich is another statement of path independence of conservative forces:

0anyclosedpath

U ds∇ • =∫r rÑ

culture

slide 11Physics 1401 - L 9 Frank Sciulli

Potential Energy Function- General

l Graph of potential energy function allows one to determineu Forcesu Stable

equilibrium locations

u Subsequent motion after initial start

( ) ( )( ) ( )

E U x K xK x E U x

= += −

Trapped near x2

Trapped near x2 or x4

Zero force=equilibrium

xdUFdx

= −

See text discussion: sec 8-5

K(x) > 0

slide 12Physics 1401 - L 9 Frank Sciulli

Pendulum Problem

y

y L LU mgLK mvU K E

θ

θ21

2

cos(1 cos )

is consta t n

= −

= −

=

+ =

review

l Describe the motion of a pendulum

l Can be done by using forces

l Some aspects can be done by using energy

slide 13Physics 1401 - L 9 Frank Sciulli

Potential Function of Pendulum

y LL

x L L

= −

≈

= ≈

212

(1 cos )

s

S

i

all

n

mθ

θ

θ

θ

θ

x

y

x

U

x

xU mgy mgLL

mgF xL

mgU xL

= ≈

= −

≈

21

2

212

ELθθ/2Lθ 21

2

slide 14Physics 1401 - L 9 Frank Sciulli

Pendulum Potential Well

l Pendulum “trapped” between extremes determined by E (fixed)

l Kinetic Energy is difference of K(x)=E-U(x)

x

y

x

U

E

xmg mg

LU x

LF x21

2 = −≈

DEMO

slide 15Physics 1401 - L 9 Frank Sciulli

l So far, we have dealt with simple systemsu point-like objectsu symmetric objects

l Now we learn how Newton’s Laws apply to complicatedsystemsu Example of issues --- F=ma --- but for what point?u Assert the answer: Center of Mass!u Prove later and get more!!

New Stuff - Ch 9-10

Complicationsbecause most masses are not simple

l New principle: conservation of momentum

slide 16Physics 1401 - L 9 Frank Sciulli

Extended body: Center of mass obeys F=ma (like point particle)

Demosl wrench (no gravity)l pix of bat in text

l First specify the system under discussion and know its mass (m)

l Determine (calculate) the center of mass (com) within the body

l Determine the total external force, Fext, on the body

co

tom

m

exc

Fam

mga gm

Above case:

= =

=

r

rr

r r

cm motion of wrench no force.MOV

RULES

slide 17Physics 1401 - L 9 Frank Sciulli

Conclusion

l Check for your MT1 blue books at entryl See all relevant exam info at websitel Next lecture, we will continue discussing

chapters 9 and 10 u Systems of particles (macroscopic bodies)uMomentum and collisions