R2-2
Work
Work is a measure of the energy that a force puts into (+) ortakes away from (–) an object as it moves.
We will see that work is a useful way to solve problems wherethe force on an object is a known function of position.
Example: the force of an object connected to an ideal spring:
xkF
where x
is the displacement from equilibrium. (Hooke’s Law)
R2-4
Vector Dot Product
F
d
If you know lengths and angle: )cos(dFW
If you know components: zzyyxx dFdFdFW
If in the same direction: dFW
If in opposite directions: dFW
If at right angles: 0W
R2-6
Work-Energy Theorem
N e t w o r k i s d o n e o n a n o b j e c t b y t h e n e t f o r c e :
f
i
x
x
netnet dxFW
K i n e t i c e n e r g y d e f i n e d f o r a n o b j e c t :
2z
2y
2x2
12
2
1 vvvmvmK
W o r k - E n e r g y T h e o r e m : ( w i t h o u t p r o o f )
netif WKK
R2-7
Class #10Take-Away Concepts
1 . W o r k i s a m e a s u r e o f e n e r g y a d d e d t o ( + ) o r t a k e n a w a y ( – ) .
)cos(dFdFW
( c o n s t a n t f o r c e )
f
i
x
x
dxFW ( v a r i a b l e f o r c e , 1 D )
2 . V e c t o r d o t p r o d u c t d e f i n e d .
3 . K i n e t i c E n e r g y : 2z
2y
2x2
12
2
1 vvvmvmK
4 . W o r k - E n e r g y T h e o r e m : netif WKK 5 . P o s i t i v e n e t w o r k m e a n s a n o b j e c t ’ s K . E . i n c r e a s e s ( s p e e d s u p ) .6 . N e g a t i v e n e t w o r k m e a n s a n o b j e c t ’ s K . E . d e c r e a s e s ( s l o w s d o w n ) .7 . Z e r o w o r k m e a n s a n o b j e c t ’ s K . E . s t a y s c o n s t a n t ( c o n s t a n t s p e e d ) .
R2-8
Does work depend on the path?Conservative Forces
For general forces, the w ork does depend on the path that w e take.H ow ever, there are som e forces for w hich w ork does not depend onthe path taken betw een the beginning and ending poin ts.T hese are called conservative forces .
A m athem atically equivalent w ay to put th is is that the w ork done bya conservative force along any closed path is exactly zero.
0xdFcons
(The funny in tegral sym bol m eans a path that closes back on itself.)
R2-9
Conservative Forces Non-Conservative Forces
Examples of Conservative Forces: Gravity Ideal Spring (Hooke’s Law) Electrostatic Force (later in Physics 1)
Examples of Non-Conservative Forces: Human Pushes and Pulls Friction
R2-10
Conservative Forcesand Potential Energy
If we are dealing with a conservative force, we can simplify the processof calculating work by introducing potential energy.
1. Define a point where the potential energy is zero (our choice).
2. Find the work done from that point to any other point in space.
(This is not too hard for most conservative forces.)
3. Define the potential energy at each point as negative the work donefrom the reference point to there. Call this function U.
4 The work done by the conservative force from any point A to anypoint B is then simply W = U(A)–U(B).
R2-11
Two Common Potential Energy Functions in Physics 1
G r a v i t a t i o n a l P o t e n t i a l E n e r g y
hgm)yy(gmU 0g ( y 0 i s o u r c h o i c e t o m a k e t h e p r o b l e m e a s i e r )
S p r i n g P o t e n t i a l E n e r g y2
02
1s )xx(kU
( x 0 i s t h e e q u i l i b r i u m p o s i t i o n a n d k i s t h e s p r i n g c o n s t a n t )
R2-12
Potential Energy, Kinetic Energy, and Conservation of Energy
R e c a l l t h e W o r k - E n e r g y T h e o r e m :
netWK A n d f o r c o n s e r v a t i v e f o r c e s w e h a v e
UW cons I f t h e n o n - c o n s e r v a t i v e f o r c e s a r e z e r o o r n e g l i g i b l e , t h e n
consnet WW P u t t i n g i t t o g e t h e r ,
UK o r 0UK A n o t h e r w a y t o s a y t h i s i s t h e t o t a l e n e r g y , K + U , i s c o n s e r v e d .
R2-13
Example ProblemSkateboarder Going Up a Ramp
hd
v
22
1 vmK 0U
0K hgmU
hgm00vm 22
1
g2v
h2
)sin(g2
v)sin(
hd
2
R2-14
Class #11Take-Away Concepts
1 . M u l t i - d i m e n s i o n a l f o r m o f w o r k i n t e g r a l :
f
i
x
x
xdFW
2 . C o n s e r v a t i v e f o r c e = w o r k d o e s n ’ t d e p e n d o n p a t h .3 . P o t e n t i a l E n e r g y d e f i n e d f o r a c o n s e r v a t i v e f o r c e :
A
0
xdF)A(U
4 . G r a v i t y : hgm)yy(gmU 0g 5 . S p r i n g : 2
02
1s )xx(kU
6 . C o n s e r v a t i o n o f e n e r g y i f o n l y c o n s e r v a t i v e f o r c e s o p e r a t e :UK o r 0UK
R2-15
Is Mechanical EnergyAlways Conserved?
T o t a l M e c h a n i c a l E n e r g y
UKE 0UKE i f o n l y c o n s e r v a t i v e f o r c e s a c t
W h e n N o n - C o n s e r v a t i v e F o r c e s A c t
consnonWUKE T h i s i s e q u i v a l e n t t o
consnoniiff WUKUK N o n - c o n s e r v a t i v e f o r c e s a d d ( + ) o r s u b t r a c t ( – ) e n e r g y .
R2-16
Example of Energy Lost to Friction (Non-Conservative Force)
hd
v
0K f J10295.18.970hgmU f
J22406470vmK2
122
1i 0Ui
A skateboarder with mass = 70 kg starts up a30º incline going 8 m/s. He goes 3 m along theincline and comes to a temporary stop. Whatwas the average force of friction (magnitude)?
m5.13)sin(dh21
R2-17
Example of Energy Lost to Friction (Non-Conservative Force)
hd
v
0K f J10295.18.970hgmU f
J22406470vmK2
122
1i 0Ui
J121122401029UKUKW iifffriction
N7.4033
1211d
WF friction
avg,friction
What do the – signs mean?
R2-18
Elastic and Inelastic Collisions
Momentum is conserved when the external forces are zero or sosmall they can be neglected during the collision. This is often true.
In many collisions a large percentage of the kinetic energy is lost.These are known as inelastic collisions. For example, any collisionin which two objects stick together is always inelastic.
If the kinetic energy after a collision is the same as before, then wehave an elastic collision. During the collision, some of the kineticenergy can convert to potential energy of various kinds, but after thecollision is over all of the kinetic energy is restored.
R2-19
Elastic Collisions inOne Dimension
+X
v2fv1f
v1i v2iI n i t i a l
F i n a l
C o n s e r v a t i o n o f M o m e n t u m :
f22f11i22i11 vmvmvmvm C o n s e r v a t i o n o f E n e r g y :
f22
22
1f1
212
1i2
222
1i1
212
1 vmvmvmvm
T w o e q u a t i o n s , t w o u n k n o w n s ( f i n a l v e l o c i t i e s ) .
R2-20
Elastic Collisions inOne Dimension
+X
v2fv1f
v1i v2iI n i t i a l
F i n a l
i221
2i1
21
21f1 v
mmm2
vmmmm
v
i221
12i1
21
1f2 v
mmmm
vmm
m2v
R2-21
Elastic Collisions inOne Dimension - Example
+X
v2fv1f
v2i
21 m2m 0v i1
1v i2 m / s
32
)1(12
2)0(
1212
v f1
m / s
31
)1(12
21)0(
122
v f2
m / s
Initial: m2 has all (–1) of themom. and KE.
Final: m1 has –4/3 of the mom.and 8/9 of the KE.m2 has +1/3 of the mom.and 1/9 of the KE.
R2-22
Class #12Take-Away Concepts
1 . M o d i f i c a t i o n o f e n e r g y c o n s e r v a t i o n i n c l u d i n g n o n -c o n s e r v a t i v e f o r c e s :
consnonWUKE 2 . N o n - c o n s e r v a t i v e w o r k a d d s ( + ) o r s u b t r a c t s ( – )
e n e r g y f r o m t h e s y s t e m .3 . E l a s t i c c o l l i s i o n p r e s e r v e s K E b e f o r e a n d a f t e r .
( D o n ’ t a s s u m e a l l c o l l i s i o n s a r e e l a s t i c , m o s t a r e n o t . )4 . S p e c i a l e q u a t i o n s f o r 1 D e l a s t i c c o l l i s i o n s .
i221
2i1
21
21f1 v
mm
m2v
mm
mmv
i221
12i1
21
1f2 v
mmmm
vmm
m2v
R2-23
Definitions
A n g u l a r P o s i t i o n :
( i n r a d i a n s )
A n g u l a r D i s p l a c e m e n t : 0
A v e r a g e o r m e a n a n g u l a r v e l o c i t y i s d e f i n e d a s f o l l o w s :
ttt 0
0avg
I n s t a n t a n e o u s a n g u l a r v e l o c i t y o r j u s t “ a n g u l a r v e l o c i t y ” :
tdd
tlim
0t
W a i t a m i n u t e ! H o w c a n a n a n g l e h a v e a v e c t o r d i r e c t i o n ?
R2-24
Direction of Angular Displacement and Angular Velocity
•Use your right hand.
•Curl your fingers in the direction of the rotation.
•Out-stretched thumb points in the direction of the angular velocity.
R2-25
Angular Acceleration
A v e r a g e a n g u l a r a c c e l e r a t i o n i s d e f i n e d a s f o l l o w s :
ttt 0
0avg
I n s t a n t a n e o u s a n g u l a r a c c e l e r a t i o n o r j u s t “ a n g u l a r a c c e l e r a t i o n ” :
2
2
0t tdd
tdd
tlim
T h e e a s i e s t w a y t o g e t t h e d i r e c t i o n o f t h e a n g u l a r a c c e l e r a t i o n i st o d e t e r m i n e t h e d i r e c t i o n o f t h e a n g u l a r v e l o c i t y a n d t h e n … I f t h e o b j e c t i s s p e e d i n g u p , a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n
a r e i n t h e s a m e d i r e c t i o n . I f t h e o b j e c t i s s l o w i n g d o w n , a n g u l a r v e l o c i t y a n d a c c e l e r a t i o n
a r e i n o p p o s i t e d i r e c t i o n s .
R2-26
Equations for Constant
1 . 00 tt
2 . 202
1000 )tt()tt(
3 . )tt)(( 002
10
4 . 202
100 )tt()tt(
5 . 020
2 2
xva
R2-27
Relationships AmongLinear and Angular Variables
MUST express angles in radians.rs rv
ra tangential
rr
rr
va 2
222
lcentripeta
T he rad ia l d irec tion is defined to be +outw ard fro m the center.
lcentripetaradial aa
R2-28
Energy in Rotation
C o n s i d e r t h e k i n e t i c e n e r g y i n a r o t a t i n g o b j e c t . T h e c e n t e r o fm a s s o f t h e o b j e c t i s n o t m o v i n g , b u t e a c h p a r t i c l e ( a t o m ) i n t h eo b j e c t i s m o v i n g a t t h e s a m e a n g u l a r v e l o c i t y ( ) .
2ii
22
12i
2i2
12ii2
1 rmrmvmK
T h e s u m m a t i o n i n t h e f i n a l e x p r e s s i o n o c c u r s o f t e n w h e na n a l y z i n g r o t a t i o n a l m o t i o n . I t i s c a l l e d t h e m o m e n t o f i n e r t i a .
R2-29
Moment of Inertia
F o r a s y s t e m o f d i s c r e t e “ p o i n t ” o b j e c t s :
2ii rmI
F o r a s o l i d o b j e c t , u s e a n i n t e g r a l w h e r e i s t h e d e n s i t y :
dzdydxrI 2
W e m a y a s k y o u t o c a l c u l a t e t h e m o m e n t o f i n e r t i a f o r p o i n t o b j e c t s , b u t w e w i l lg i v e y o u a f o r m u l a f o r a s o l i d o b j e c t o r j u s t g i v e y o u i t s m o m e n t o f i n e r t i a .
I for a solid sphere: 25
2 RMII for a spherical shell: 2
3
2 RMI
R2-30
Correspondence BetweenLinear and Rotational Motion
xvaIm F
22
1 IK
I
You will solve many rotation problemsusing exactly the same techniques youlearned for linear motion problems.
R2-31
Class #13Take-Away Concepts
1 . D e f i n i t i o n s o f r o t a t io n a l q u a n t i t i e s : , , .
2 . C e n t r ip e t a l a n d t a n g e n t i a l a c c e l e r a t io n .
3 . M o m e n t o f i n e r t i a : 2
ii rmI 4 . R o ta t io n a l k in e t i c e n e r g y : 2
2
1 IK 5 . I n t r o d u c t io n to to r q u e :
I6 . C o r r e s p o n d e n c e
x v aIm F
R2-32
Review of Torque
For linear motion, we have “F = m a”. For rotation, we have
I
The symbol “” is torque. We will define it more precisely today.
When the rotation is speeding up, and are in thesame direction. When the rotation is slowing down, and are inopposite directions.
Torque and angular acceleration arealways in the same direction in Physics 1.
R2-33
The Vector Cross Product
We learned how to “multiply” two vectors to get a scalar.That was the “dot” product:
)cos(|b||a|bad
Now we will “multiply” two vectors to get another vector:
bac; )sin(|b||a||c|
The direction comes from the right-hand rule. It is at a right angle
to the plane formed by a and b. In other words, the cross product
is at right angles to both a and b. (3D thinking required!)
R2-35
Torque as a Cross Product
Fr
)sin(|F||r|||
r
i s t h e v e c t o r f r o m t h e a x i s o f r o t a t i o n t o w h e r e t h e f o r c e i s a p p l i e d .
T h e t o r q u e c a n b e z e r o i n t h r e e d i f f e r e n t w a y s :
1 . N o f o r c e i s a p p l i e d ( 0|F|
) .2 . T h e f o r c e i s a p p l i e d a t t h e a x i s o f r o t a t i o n ( 0|r|
) .
3 . F
a n d r
i n t h e s a m e o r o p p o s i t e d i r e c t i o n s ( 0)sin( ) .
R2-36
Angular Momentum of a Particle
vmp
r
center of rotation (defined) A n g u la r m o m e n t u m o f a p a r t i c l eo n c e a c e n t e r i s d e f i n e d :
pr
l
( W h a t i s t h e d i r e c t i o n o f a n g u l a rm o m e n t u m h e r e ? )
Once we define a center (or axis) of rotation, any object with alinear momentum that does not move directly through that pointhas an angular momentum defined relative to the chosen center.
R2-37
Class #14Take-Away Concepts
1. : Speeding up, slowing down. I .
2. Definition of vector cross product:
bac ; )sin(|b||a||c|
3. Torque as a cross product: Fr .
4. Angular momentum of a particle: pr
l .
R2-38
How Does Angular Momentum of a Particle Change with Time?
Take the time derivative of angular momentum:
tdpd
rptdrd
)pr(td
dtd
d
l
Find each term separately:
0pvptdrd
(Why?)
netnetFrtdpd
r (Why?)
so
nettdd l
(Newton’s 2nd Law for angular momentum.)
R2-39
Angular Momentum of a Particle:Does It Change if = 0?
vmp = 1 kg m/s (+X dir.)(0,0)
(0,–3) (4,–3)
X
Y
The figure at the left shows the sameparticle at two different times. No forces(or torques) act on the particle.Is its angular momentum constant?(Check magnitudes at the two times.)
Blue angle: = 90ºl = r p sin() = (3) (1) sin(90º) = 3 kg m2/s
Red angle: = arctan(3/4) = 36.87ºl = r p sin() = (5) (1) sin(36.87º) = 3 kg m2/s
[r sin()] is the component of r at a right angle to p
. It is
constant.It is also the distance at closest approach to the center.
r
(blue)
r(red)
R2-40
Conservation of Angular Momentum
Take (for example) two rotating objects that interact.
1onext2from1on1
tdd l
2onext1from2on2
tdd l
The total angular momentum is the sum of 1 and 2:
2onext1onext21
tdd
tdd
tdLd
ll
(Why?)
If there are no external torques, then
0tdLd
R2-41
Class #15Take-Away Concepts
1. Angular momentum of a particle (review): pr
l .2. Newton’s 2nd Law for angular momentum:
nettdd l
3. Conservation of angular momentum (no ext. torque):
0tdLd
R2-42
Formula Sheet Organization
L i n e a r K i n e m a t i c s
1 . 00 ttavv
2 . 202
1000 )tt(a)tt(vxx
3 . )tt)(vv(xx 002
10
4 . 202
100 )tt(a)tt(vxx
5 . 020
2 xxa2vv
R2-43
Formula Sheet OrganizationN e w t o n ’ s 2 n d L a w a n d L i n e a r M o m e n t u m
6 . amFF net
1 0 . vmp
1 1 .td
pdFF net
1 2 . pdtFJ
1 3 . ipP
1 4 . extFtd
Pd
1 5 . imM
1 6 . iicm xmM
1x iicm ym
M
1y
1 7 . cmvMP
R2-44
Formula Sheet Organization
W o r k a n d E n e r g y ( L i n e a r M o t i o n )
1 8 . yyxx baba)cos(baba
1 9 . dFW
2 0 . xdFW
2 1 . )vv(mvmK 2y
2x2
12
2
1
2 2 . netif WKK
2 3 . xdFU cons
2 4 . )yy(gmU 0g
2 5 . 202
1s )xx(kU
2 6 . consnonWUK
R2-45
Formula Sheet Organization
R o t a t i o n a l K i n e m a t i c s
3 0 . 00 tt
3 1 . 202
1000 )tt()tt(
3 2 . )tt)(( 002
10
3 3 . 202
100 )tt()tt(
3 4 . 020
2 2
R2-46
Formula Sheet OrganizationR o t a t io n a l M o t io n / L in e a r M o t io n
7 .v
r2T
8 . rr
va 2
2
lcentripeta
9 . rmr
vmF 2
2
lcentripeta
2 7 . rs 2 8 . rv gentialtan
2 9 . ra gentialtan
R2-47
Formula Sheet OrganizationNewton’s 2nd Law and Angular Mom.
35. )sin(baba
36. 2ii rmI
39. Fr
40.td
LdI
41. pr
l
42. iL l
43.
IL