Physics 121,Lecture 12,Pg 1
Physics 121,Sections 9,10,11,and 12
Lecture 12
Today’s Topics:
? Homework 5,Due Friday Oct,7 @ 6:00PM
? Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Review session,Friday Oct,7 @ 11:00AM in IMS-20
? Chapter 6,Work and Energy
? Forms of energy
? Definition of work
? Work-kinetic energy theorem
? Examples
Physics 121,Lecture 12,Pg 2
Summary
(with comparison to 1-D kinematics)
Angular Linear
c o n s ta n t??
??
? ? ? 0 ??t
? ? ? ?? ? ?0 0 212t t
co n st a n ta ?
v v at? ?0
x x v t at? ? ?0 0 212
And for a point at a distance R from the rotation axis:
x = R????????????v = ?R ??????????a = ?R
Physics 121,Lecture 12,Pg 3
Work & Energy
? One of the most important concepts in physics.
?Alternative approach to mechanics.
? Many applications beyond mechanics.
?Thermodynamics (movement of heat).
?Quantum mechanics...
? Very useful tools.
?You will learn new (sometimes much easier) ways to
solve problems.
Physics 121,Lecture 12,Pg 4
Forms of Energy
? Kinetic,Energy of motion.
?A car on the highway has kinetic energy.
?We have to remove this energy to stop it.
?The breaks of a car get HOT !
?This is an example of turning one form of energy into
another (thermal energy).
?Kinetic energy is given by,K = (1/2) mv2
Physics 121,Lecture 12,Pg 5
Energy Conservation
? Energy cannot be destroyed or created.
?Just changed from one form to another.
? We say energy is conserved !
?True for any isolated system.
?i.e when we put on the brakes,the kinetic energy of the car
is turned into heat using friction in the brakes,The total
energy of the,car-breaks-road-atmosphere” system is the
same.
?The energy of the car,alone” is not conserved...
? It is reduced by the braking.
? Doing,work” on an otherwise isolated system will change it’s
“energy”...
Physics 121,Lecture 12,Pg 6
Definition of Work:
Ingredients,Force ( F ),displacement ( ? r )
Work,W,of a constant force F
acting through a displacement ? r is:
W = F ? r cos ? ?
F
? rF
r
Physics 121,Lecture 12,Pg 7
Definition of Work...
? Only the component of F along the displacement
is doing work.
?Example,Train on a track.
? r
F
?
F cos ?
Physics 121,Lecture 12,Pg 8
Lecture 12,ACT 1
Work
? A box is pulled up a rough (m > 0) incline by a rope-pulley-
weight arrangement as shown below.
?How many forces are doing work on the box?
(a) 2
(b) 3
(c) 4
Physics 121,Lecture 12,Pg 9
Work,1-D Example
(constant force)
? A force F = 10N pushes a box across a frictionless
floor for a distance ?x = 5m.
?x
F
Work done by F on box,
WF = F ·?x = F?x (since F is parallel to ?x)
WF = (10 N)x(5m) = 50 N-m.
Physics 121,Lecture 12,Pg 10
Units:
N-m (Joule) Dyne-cm (erg)
= 10-7 J
BTU = 1054 J
calorie = 4.184 J
foot-lb = 1.356 J
eV = 1.6x10-19 J
cgs othermks
Force x Distance = Work
Newton x
[M][L] / [T]2
Meter = Joule
[L] [M][L]2 / [T]2
Physics 121,Lecture 12,Pg 11
Work & Kinetic Energy:
? A force F = 10N pushes a box across a frictionless
floor for a distance ?x = 5m,The speed of the box is v1
before the push,and v2 after the push.
?x
F
v1 v2
i
m
Physics 121,Lecture 12,Pg 12
Work & Kinetic Energy...
? Since the force F is constant,acceleration a will be
constant,We have shown that for constant a:
?v22 - v12 = 2a(x2-x1 ) = 2a?x.
?multiply by 1/2m,1/2mv22 - 1/2mv12 = ma ?x
?But F = ma 1/2mv22 - 1/2mv12 = F?x
?x
F
v1 v2
a
i
m
Physics 121,Lecture 12,Pg 13
Work & Kinetic Energy...
? So we find that
?1/2mv22 - 1/2mv12 = F?x = WF
? Define Kinetic Energy KE,KE = 1/2mv2
?KE2 - KE1 = WF
?WF = ?KE (Work kinetic-energy theorem)
?x
F
v1 v2
a
i
m
Physics 121,Lecture 12,Pg 14
Work Kinetic-Energy Theorem:
{Net Work done on object}
=
{change in kinetic energy of object}
KW net ??
12 KK ??
2122 mv21mv21 ??
Physics 121,Lecture 12,Pg 15
Lecture 12,ACT 2
Kinetic Energy
? To practice your pitching you use two baseballs,The first time
you throw a slow curve and clock the speed at 50 mph (~25
m/s),The second time you go with high heat and the radar
gun clocks the pitch at 100 mph,What is the ratio of the
kinetic energy of the fast ball versus the curve ball?
(a) 1/4 (b) 1/2 (c) 1 (d) 2 (e) 4
Physics 121,Lecture 12,Pg 16
A simple application:
Work done by gravity on a falling object
? What is the speed of an object after falling a distance H,
assuming it starts at rest?
? Wg = mg ? r cos(0) = mgH
Wg = mgH
Work Kinetic-Energy Theorem:
Wg = mgH = 1/2mv2
? r mg
H
j
v0 = 0
vv gH? 2
Physics 121,Lecture 12,Pg 17
What about a sum of forces?
Suppose FTOT = F1 + F2 and the
displacement is ? r.
The work done by each force is:
W1 = F1 cos ?1 ? r,and
W2 = F2 cos ?2? r
WTOT = W1 + W2
WTOT = FTOTcos ?? r
It’s the total force that matters !!
FTOT
? rF1
F2
??1
?2
Physics 121,Lecture 12,Pg 18
Comments:
? Time interval not relevant.
?Run up the stairs quickly or slowly...same W.
Since W = F cos ? ? r
? No work is done if:
?F = 0 or ? r = 0 or ? = 90o
Physics 121,Lecture 12,Pg 19
Comments...
W = F cos ? ?r
? No work done if ? = 90o.
?No work done by T.
?No work done by N.
T
v
v
N
Physics 121,Lecture 12,Pg 20
Work & Energy
? An inclined plane is accelerating with constant acceleration a,
A box resting on the plane is held in place by static friction,
How many forces are doing work on the block?
a
(a) 1 (b) 2 (c) 3
Physics 121,Lecture 12,Pg 21
Lecture 12,ACT 3
Work & Energy
? Two blocks having mass m1 and m2 where m1 > m2,They are
sliding on a frictionless floor and have the same kinetic energy
when they encounter a long rough stretch (i.e,m > 0) which
slows them down to a stop.
Which one will go farther before stopping?
(a) m1 (b) m2 (c) they will go the same distance
m1
m2
Physics 121,Lecture 12,Pg 22
Lecture 12,ACT 4
Work & Energy
? You like to drive home fast,slam on your brakes at the bottom
of the driveway,and screech to a stop laying rubber all the
way,It’s particularly fun when your mother is in the car with
you,You practice this trick driving at 20 mph and with some
groceries in your car with the same mass as your mama,You
find that you only travel half way up the driveway,Thus when
your mom joins you in the car,you try it driving twice as fast,
How far will you go this time?
(a) The same distance,Not so exciting.
(b) ? 2 times as far (only 7/10 of the way up the driveway)
(c) twice as far,right to the door,Whoopee!
(d) four times as far,Crashes into house,Sorry Ma.
Physics 121,Lecture 12,Pg 23
Work done by gravity:
? Wg = mg?r cos ?
= -mg?y
Wg = -mg?y
Depends only on ?y !
not on path taken!
j
m
?r
mg
??y
?
m
Physics 121,Lecture 12,Pg 24
Lecture 12,ACT 5
Falling Objects
? Three objects of mass m begin at height h with velocity 0,
One falls straight down,one slides down a frictionless
inclined plane,and one swings on the end of a pendulum,
What is the relationship between their velocities when they
have fallen to height 0?
(a) vf > vi > vp (b) vf > vp > vi (c) vf = vp = vi
v=0
vi
H
v=0
vp
v=0
vf
Free Fall Frictionless incline Pendulum
Physics 121,Lecture 12,Pg 25
Lifting a book with your hand:
What is the total work done on the book
? First calculate the work done by gravity:
Wg = mg ?r cos ? = -mg ?r
? Now find the work done by
the hand:
WHAND = FHAND?r cos ? = FHAND ?r mg
?r FHAND
v = const
a = 0
Physics 121,Lecture 12,Pg 26
Example,Lifting a book...
Wg = -mg ?r
WHAND = FHAND ?r
WTOT = WHAND + Wg
= FHAND ?r -mg ?r
= (FHAND -mg) ?r
= 0 since 1/2 KE = 0 (v= const)
? So WTOT = 0 !!
mg
?r FHAND
v = const
a = 0
Physics 121,Lecture 12,Pg 27
Example,Lifting a book...
?Work Kinetic-Energy Theorem says,W = ?KE
{Net Work done on object}=
{change in kinetic energy of object}
In this case,v is constant
so ?KE = 0 and so W must
be 0,as we found.
mg
?r FHAND
v = const
a = 0
Physics 121,Lecture 12,Pg 28
Work done by Variable Force,(1D)
? When the force was constant,
we wrote W = F?x
?area under F vs x plot:
F
x
Wg
?x
F(x)
x1 x2 ?x ??
W ? F ( x )? x?
Physics 121,Lecture 12,Pg 29
Springs
? A very common problem with a variable force is a spring,
? In this spring,the force gets greater as the spring is further
compressed,
? Hook’s Law,
FS = - k ?x
?x is the amount the spring is stretched or compressed
from it resting position.
F
?x
Active Figure
Physics 121,Lecture 12,Pg 30
1-D Variable Force Example,Spring
? For a spring we know that Fx = -kx (Hook’s law).
F(x) x2
x
x1
-kxrelaxed position
F= - k x1
F= - k x2
Physics 121,Lecture 12,Pg 31
Spring...
? The work done by the spring Ws during a displacement
from x1 to x2 is the area under the F(x) vs x plot
between x1 and x2.
Ws
F(x) x2
x
x1
-kxrelaxed position
Physics 121,Lecture 12,Pg 32
Spring...
??
W s ? ? 12 k x 22 ? x 12? ?
? The work done by the spring Ws during a
displacement from x1 to x2 is the area under the
F(x) vs x plot between x1 and x2.
x2 x1
F(x)
x
Ws
kx1
kx2
-kx
Ws = - 1/2 [ ( kx2) (x2) - (kx1) (x1) ]
Physics 121,Lecture 12,Pg 33
Lecture 12,ACT 6
Work & Energy
? A box sliding on a horizontal frictionless surface runs into a
fixed spring,compressing it a distance x from its relaxed
position while momentarily coming to rest.
?If the initial speed of the box were doubled and its mass
were halved,how far x’ would the spring compress?
x
(a) (b) (c)x'x = x2'x = x2'x =
Physics 121,Lecture 12,Pg 34
Problem,Spring pulls on mass.
? A spring (constant k) is stretched a distance d,and a mass m
is hooked to its end,The mass is released (from rest),What
is the speed of the mass when it returns to the relaxed
position if it slides without friction?
relaxed position
stretched position (at rest)
d
after release
back at relaxed position
vr
v
m
m
m
m
Physics 121,Lecture 12,Pg 35
Problem,Spring pulls on mass.
? First find the net work done on the mass during the motion
from x= d to x= 0 (only due to the spring):
stretched position (at rest)
d
relaxed position
vr
m
m
i
??
W s ? ? 12 k x 22 ? x 12? ? ? ? 12 k 0 2 ? d 2? ? ? 12 kd 2
Physics 121,Lecture 12,Pg 36
Problem,Spring pulls on mass.
? Now find the change in kinetic energy of the mass:
stretched position (at rest)
d
relaxed position
vr
m
m
i
??
? KE ? 12 mv 22 ? 12 mv 12 ? 12 mv r2
Physics 121,Lecture 12,Pg 37
Problem,Spring pulls on mass.
? Now use work kinetic-energy theorem,
Wnet = WS = ? KE.
stretched position (at rest)
d
relaxed position
vr
m
m
i
1
2
2kd ?2
rmv21 m
kdv
r ?
Physics 121,Lecture 12,Pg 38
Problem,Spring pulls on mass.
? Now suppose there is a coefficient of friction m between the
block and the floor
? The total work done on the block is now the sum of the work
done by the spring WS (same as before) and the work done
by friction Wf.
Wf = f,?r = - m mg d
stretched position (at rest)
d
relaxed position
vr
m
m
i
f = m mg
?r
Physics 121,Lecture 12,Pg 39
Problem,Spring pulls on mass.
? Again use Wnet = WS + Wf = ?KE
Wf = - mmg d
stretched position (at rest)
d
relaxed position
vr
m
m
i
f = mmg
?r
W kdS ? 12 2 2
rmv2
1K ??
2r2 mv21m gdkd21 ?? m
??
v r ? km d 2 ? 2 m gd
Physics 121,Lecture 12,Pg 40
Conservative Forces:
? In general,if the work done does not depend on the
path taken,the force involved is said to be
conservative.
? Gravity is a conservative force:
? Gravity near the Earth’s surface:
? A spring produces a conservative force:
??
W s ? ? 12 k x 22 ? x 12? ?
ymgW g ???
??
?
??
? ??
12
g R
1
R
1G M mW
Physics 121,Lecture 12,Pg 41
Conservative Forces:
? We have seen that the work done by a conservative force
does not depend on the path taken.
W1
The work done can be,reclaimed”.
W2
W1
W2
W1 = W2
WNET = W1 - W2 = 0
Therefore the work done in
a closed path is 0.
Physics 121,Lecture 12,Pg 42
Potential Energy
? For any conservative force F we can define a
potential energy function U in the following way:
?The work done by a conservative force is equal and
opposite to the change in the potential energy
function.
? This can be written as:
W = - ?U
?U = U2 - U1 = - W r1
r2 U2
U1
Physics 121,Lecture 12,Pg 43
Gravitational Potential Energy
? We have seen that the work done by gravity near
the Earth’s surface when an object of mass m is
lifted a distance ?y is
Wg = -mg?y.
? The change in potential energy of this object is
therefore:
?U = - Wg = mg?y.
?y
m
Wg = -mg?y
j
Physics 121,Lecture 12,Pg 44
Gravitational Potential Energy
? So we see that the change in U near the earths surface is:
?U = - Wg = mg?y = mg(y2 -y1).
? So U = mgy + U0 where U0 is an arbitrary constant.
? Having an arbitrary constant U0 is equivalent to saying that
we can choose the y location where U = 0 to be anywhere
we want to.
y1
m
Wg = -mg?y
j y2
Physics 121,Lecture 12,Pg 45
Potential Energy Recap:
? For any conservative force we can define a potential
energy function U such that:
? The potential energy function U is always defined
only up to an additive constant.
?You can choose the location where U = 0 to be
anywhere convenient.
?U = U2 - U1 = - W
Physics 121,Lecture 12,Pg 46
Conservative Forces & Potential Energies
(stuff you should know):
Force
F
Work
W(1-2)
Change in P.E
?U = U2 - U1
P.E,function
U
Fg = -mg j
Fg = r
Fs = -kx
^
^
??
?
??
? ?
12 R
1
R
1G M m
? ?? ?12 22 12k x x
-mg(y2-y1) mg(y2-y1)
? ???
?
?
?
?
G M m
R R
1 1
2 1
? ?12 22 12k x x?
? G M mR 2
mgy + C
CRGMm ??
Ckx ?221
Physics 121,Lecture 12,Pg 47
Recap of today’s lecture
? Homework 5,Due Friday Oct,7 @ 6:00PM
?Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Review session,Friday Oct,7 @ 11:00AM in IMS-20
? Chapter 6,Work and Energy
?Forms of energy
?Definition of work
?Work-kinetic energy theorem
?Examples
Physics 121,Sections 9,10,11,and 12
Lecture 12
Today’s Topics:
? Homework 5,Due Friday Oct,7 @ 6:00PM
? Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Review session,Friday Oct,7 @ 11:00AM in IMS-20
? Chapter 6,Work and Energy
? Forms of energy
? Definition of work
? Work-kinetic energy theorem
? Examples
Physics 121,Lecture 12,Pg 2
Summary
(with comparison to 1-D kinematics)
Angular Linear
c o n s ta n t??
??
? ? ? 0 ??t
? ? ? ?? ? ?0 0 212t t
co n st a n ta ?
v v at? ?0
x x v t at? ? ?0 0 212
And for a point at a distance R from the rotation axis:
x = R????????????v = ?R ??????????a = ?R
Physics 121,Lecture 12,Pg 3
Work & Energy
? One of the most important concepts in physics.
?Alternative approach to mechanics.
? Many applications beyond mechanics.
?Thermodynamics (movement of heat).
?Quantum mechanics...
? Very useful tools.
?You will learn new (sometimes much easier) ways to
solve problems.
Physics 121,Lecture 12,Pg 4
Forms of Energy
? Kinetic,Energy of motion.
?A car on the highway has kinetic energy.
?We have to remove this energy to stop it.
?The breaks of a car get HOT !
?This is an example of turning one form of energy into
another (thermal energy).
?Kinetic energy is given by,K = (1/2) mv2
Physics 121,Lecture 12,Pg 5
Energy Conservation
? Energy cannot be destroyed or created.
?Just changed from one form to another.
? We say energy is conserved !
?True for any isolated system.
?i.e when we put on the brakes,the kinetic energy of the car
is turned into heat using friction in the brakes,The total
energy of the,car-breaks-road-atmosphere” system is the
same.
?The energy of the car,alone” is not conserved...
? It is reduced by the braking.
? Doing,work” on an otherwise isolated system will change it’s
“energy”...
Physics 121,Lecture 12,Pg 6
Definition of Work:
Ingredients,Force ( F ),displacement ( ? r )
Work,W,of a constant force F
acting through a displacement ? r is:
W = F ? r cos ? ?
F
? rF
r
Physics 121,Lecture 12,Pg 7
Definition of Work...
? Only the component of F along the displacement
is doing work.
?Example,Train on a track.
? r
F
?
F cos ?
Physics 121,Lecture 12,Pg 8
Lecture 12,ACT 1
Work
? A box is pulled up a rough (m > 0) incline by a rope-pulley-
weight arrangement as shown below.
?How many forces are doing work on the box?
(a) 2
(b) 3
(c) 4
Physics 121,Lecture 12,Pg 9
Work,1-D Example
(constant force)
? A force F = 10N pushes a box across a frictionless
floor for a distance ?x = 5m.
?x
F
Work done by F on box,
WF = F ·?x = F?x (since F is parallel to ?x)
WF = (10 N)x(5m) = 50 N-m.
Physics 121,Lecture 12,Pg 10
Units:
N-m (Joule) Dyne-cm (erg)
= 10-7 J
BTU = 1054 J
calorie = 4.184 J
foot-lb = 1.356 J
eV = 1.6x10-19 J
cgs othermks
Force x Distance = Work
Newton x
[M][L] / [T]2
Meter = Joule
[L] [M][L]2 / [T]2
Physics 121,Lecture 12,Pg 11
Work & Kinetic Energy:
? A force F = 10N pushes a box across a frictionless
floor for a distance ?x = 5m,The speed of the box is v1
before the push,and v2 after the push.
?x
F
v1 v2
i
m
Physics 121,Lecture 12,Pg 12
Work & Kinetic Energy...
? Since the force F is constant,acceleration a will be
constant,We have shown that for constant a:
?v22 - v12 = 2a(x2-x1 ) = 2a?x.
?multiply by 1/2m,1/2mv22 - 1/2mv12 = ma ?x
?But F = ma 1/2mv22 - 1/2mv12 = F?x
?x
F
v1 v2
a
i
m
Physics 121,Lecture 12,Pg 13
Work & Kinetic Energy...
? So we find that
?1/2mv22 - 1/2mv12 = F?x = WF
? Define Kinetic Energy KE,KE = 1/2mv2
?KE2 - KE1 = WF
?WF = ?KE (Work kinetic-energy theorem)
?x
F
v1 v2
a
i
m
Physics 121,Lecture 12,Pg 14
Work Kinetic-Energy Theorem:
{Net Work done on object}
=
{change in kinetic energy of object}
KW net ??
12 KK ??
2122 mv21mv21 ??
Physics 121,Lecture 12,Pg 15
Lecture 12,ACT 2
Kinetic Energy
? To practice your pitching you use two baseballs,The first time
you throw a slow curve and clock the speed at 50 mph (~25
m/s),The second time you go with high heat and the radar
gun clocks the pitch at 100 mph,What is the ratio of the
kinetic energy of the fast ball versus the curve ball?
(a) 1/4 (b) 1/2 (c) 1 (d) 2 (e) 4
Physics 121,Lecture 12,Pg 16
A simple application:
Work done by gravity on a falling object
? What is the speed of an object after falling a distance H,
assuming it starts at rest?
? Wg = mg ? r cos(0) = mgH
Wg = mgH
Work Kinetic-Energy Theorem:
Wg = mgH = 1/2mv2
? r mg
H
j
v0 = 0
vv gH? 2
Physics 121,Lecture 12,Pg 17
What about a sum of forces?
Suppose FTOT = F1 + F2 and the
displacement is ? r.
The work done by each force is:
W1 = F1 cos ?1 ? r,and
W2 = F2 cos ?2? r
WTOT = W1 + W2
WTOT = FTOTcos ?? r
It’s the total force that matters !!
FTOT
? rF1
F2
??1
?2
Physics 121,Lecture 12,Pg 18
Comments:
? Time interval not relevant.
?Run up the stairs quickly or slowly...same W.
Since W = F cos ? ? r
? No work is done if:
?F = 0 or ? r = 0 or ? = 90o
Physics 121,Lecture 12,Pg 19
Comments...
W = F cos ? ?r
? No work done if ? = 90o.
?No work done by T.
?No work done by N.
T
v
v
N
Physics 121,Lecture 12,Pg 20
Work & Energy
? An inclined plane is accelerating with constant acceleration a,
A box resting on the plane is held in place by static friction,
How many forces are doing work on the block?
a
(a) 1 (b) 2 (c) 3
Physics 121,Lecture 12,Pg 21
Lecture 12,ACT 3
Work & Energy
? Two blocks having mass m1 and m2 where m1 > m2,They are
sliding on a frictionless floor and have the same kinetic energy
when they encounter a long rough stretch (i.e,m > 0) which
slows them down to a stop.
Which one will go farther before stopping?
(a) m1 (b) m2 (c) they will go the same distance
m1
m2
Physics 121,Lecture 12,Pg 22
Lecture 12,ACT 4
Work & Energy
? You like to drive home fast,slam on your brakes at the bottom
of the driveway,and screech to a stop laying rubber all the
way,It’s particularly fun when your mother is in the car with
you,You practice this trick driving at 20 mph and with some
groceries in your car with the same mass as your mama,You
find that you only travel half way up the driveway,Thus when
your mom joins you in the car,you try it driving twice as fast,
How far will you go this time?
(a) The same distance,Not so exciting.
(b) ? 2 times as far (only 7/10 of the way up the driveway)
(c) twice as far,right to the door,Whoopee!
(d) four times as far,Crashes into house,Sorry Ma.
Physics 121,Lecture 12,Pg 23
Work done by gravity:
? Wg = mg?r cos ?
= -mg?y
Wg = -mg?y
Depends only on ?y !
not on path taken!
j
m
?r
mg
??y
?
m
Physics 121,Lecture 12,Pg 24
Lecture 12,ACT 5
Falling Objects
? Three objects of mass m begin at height h with velocity 0,
One falls straight down,one slides down a frictionless
inclined plane,and one swings on the end of a pendulum,
What is the relationship between their velocities when they
have fallen to height 0?
(a) vf > vi > vp (b) vf > vp > vi (c) vf = vp = vi
v=0
vi
H
v=0
vp
v=0
vf
Free Fall Frictionless incline Pendulum
Physics 121,Lecture 12,Pg 25
Lifting a book with your hand:
What is the total work done on the book
? First calculate the work done by gravity:
Wg = mg ?r cos ? = -mg ?r
? Now find the work done by
the hand:
WHAND = FHAND?r cos ? = FHAND ?r mg
?r FHAND
v = const
a = 0
Physics 121,Lecture 12,Pg 26
Example,Lifting a book...
Wg = -mg ?r
WHAND = FHAND ?r
WTOT = WHAND + Wg
= FHAND ?r -mg ?r
= (FHAND -mg) ?r
= 0 since 1/2 KE = 0 (v= const)
? So WTOT = 0 !!
mg
?r FHAND
v = const
a = 0
Physics 121,Lecture 12,Pg 27
Example,Lifting a book...
?Work Kinetic-Energy Theorem says,W = ?KE
{Net Work done on object}=
{change in kinetic energy of object}
In this case,v is constant
so ?KE = 0 and so W must
be 0,as we found.
mg
?r FHAND
v = const
a = 0
Physics 121,Lecture 12,Pg 28
Work done by Variable Force,(1D)
? When the force was constant,
we wrote W = F?x
?area under F vs x plot:
F
x
Wg
?x
F(x)
x1 x2 ?x ??
W ? F ( x )? x?
Physics 121,Lecture 12,Pg 29
Springs
? A very common problem with a variable force is a spring,
? In this spring,the force gets greater as the spring is further
compressed,
? Hook’s Law,
FS = - k ?x
?x is the amount the spring is stretched or compressed
from it resting position.
F
?x
Active Figure
Physics 121,Lecture 12,Pg 30
1-D Variable Force Example,Spring
? For a spring we know that Fx = -kx (Hook’s law).
F(x) x2
x
x1
-kxrelaxed position
F= - k x1
F= - k x2
Physics 121,Lecture 12,Pg 31
Spring...
? The work done by the spring Ws during a displacement
from x1 to x2 is the area under the F(x) vs x plot
between x1 and x2.
Ws
F(x) x2
x
x1
-kxrelaxed position
Physics 121,Lecture 12,Pg 32
Spring...
??
W s ? ? 12 k x 22 ? x 12? ?
? The work done by the spring Ws during a
displacement from x1 to x2 is the area under the
F(x) vs x plot between x1 and x2.
x2 x1
F(x)
x
Ws
kx1
kx2
-kx
Ws = - 1/2 [ ( kx2) (x2) - (kx1) (x1) ]
Physics 121,Lecture 12,Pg 33
Lecture 12,ACT 6
Work & Energy
? A box sliding on a horizontal frictionless surface runs into a
fixed spring,compressing it a distance x from its relaxed
position while momentarily coming to rest.
?If the initial speed of the box were doubled and its mass
were halved,how far x’ would the spring compress?
x
(a) (b) (c)x'x = x2'x = x2'x =
Physics 121,Lecture 12,Pg 34
Problem,Spring pulls on mass.
? A spring (constant k) is stretched a distance d,and a mass m
is hooked to its end,The mass is released (from rest),What
is the speed of the mass when it returns to the relaxed
position if it slides without friction?
relaxed position
stretched position (at rest)
d
after release
back at relaxed position
vr
v
m
m
m
m
Physics 121,Lecture 12,Pg 35
Problem,Spring pulls on mass.
? First find the net work done on the mass during the motion
from x= d to x= 0 (only due to the spring):
stretched position (at rest)
d
relaxed position
vr
m
m
i
??
W s ? ? 12 k x 22 ? x 12? ? ? ? 12 k 0 2 ? d 2? ? ? 12 kd 2
Physics 121,Lecture 12,Pg 36
Problem,Spring pulls on mass.
? Now find the change in kinetic energy of the mass:
stretched position (at rest)
d
relaxed position
vr
m
m
i
??
? KE ? 12 mv 22 ? 12 mv 12 ? 12 mv r2
Physics 121,Lecture 12,Pg 37
Problem,Spring pulls on mass.
? Now use work kinetic-energy theorem,
Wnet = WS = ? KE.
stretched position (at rest)
d
relaxed position
vr
m
m
i
1
2
2kd ?2
rmv21 m
kdv
r ?
Physics 121,Lecture 12,Pg 38
Problem,Spring pulls on mass.
? Now suppose there is a coefficient of friction m between the
block and the floor
? The total work done on the block is now the sum of the work
done by the spring WS (same as before) and the work done
by friction Wf.
Wf = f,?r = - m mg d
stretched position (at rest)
d
relaxed position
vr
m
m
i
f = m mg
?r
Physics 121,Lecture 12,Pg 39
Problem,Spring pulls on mass.
? Again use Wnet = WS + Wf = ?KE
Wf = - mmg d
stretched position (at rest)
d
relaxed position
vr
m
m
i
f = mmg
?r
W kdS ? 12 2 2
rmv2
1K ??
2r2 mv21m gdkd21 ?? m
??
v r ? km d 2 ? 2 m gd
Physics 121,Lecture 12,Pg 40
Conservative Forces:
? In general,if the work done does not depend on the
path taken,the force involved is said to be
conservative.
? Gravity is a conservative force:
? Gravity near the Earth’s surface:
? A spring produces a conservative force:
??
W s ? ? 12 k x 22 ? x 12? ?
ymgW g ???
??
?
??
? ??
12
g R
1
R
1G M mW
Physics 121,Lecture 12,Pg 41
Conservative Forces:
? We have seen that the work done by a conservative force
does not depend on the path taken.
W1
The work done can be,reclaimed”.
W2
W1
W2
W1 = W2
WNET = W1 - W2 = 0
Therefore the work done in
a closed path is 0.
Physics 121,Lecture 12,Pg 42
Potential Energy
? For any conservative force F we can define a
potential energy function U in the following way:
?The work done by a conservative force is equal and
opposite to the change in the potential energy
function.
? This can be written as:
W = - ?U
?U = U2 - U1 = - W r1
r2 U2
U1
Physics 121,Lecture 12,Pg 43
Gravitational Potential Energy
? We have seen that the work done by gravity near
the Earth’s surface when an object of mass m is
lifted a distance ?y is
Wg = -mg?y.
? The change in potential energy of this object is
therefore:
?U = - Wg = mg?y.
?y
m
Wg = -mg?y
j
Physics 121,Lecture 12,Pg 44
Gravitational Potential Energy
? So we see that the change in U near the earths surface is:
?U = - Wg = mg?y = mg(y2 -y1).
? So U = mgy + U0 where U0 is an arbitrary constant.
? Having an arbitrary constant U0 is equivalent to saying that
we can choose the y location where U = 0 to be anywhere
we want to.
y1
m
Wg = -mg?y
j y2
Physics 121,Lecture 12,Pg 45
Potential Energy Recap:
? For any conservative force we can define a potential
energy function U such that:
? The potential energy function U is always defined
only up to an additive constant.
?You can choose the location where U = 0 to be
anywhere convenient.
?U = U2 - U1 = - W
Physics 121,Lecture 12,Pg 46
Conservative Forces & Potential Energies
(stuff you should know):
Force
F
Work
W(1-2)
Change in P.E
?U = U2 - U1
P.E,function
U
Fg = -mg j
Fg = r
Fs = -kx
^
^
??
?
??
? ?
12 R
1
R
1G M m
? ?? ?12 22 12k x x
-mg(y2-y1) mg(y2-y1)
? ???
?
?
?
?
G M m
R R
1 1
2 1
? ?12 22 12k x x?
? G M mR 2
mgy + C
CRGMm ??
Ckx ?221
Physics 121,Lecture 12,Pg 47
Recap of today’s lecture
? Homework 5,Due Friday Oct,7 @ 6:00PM
?Ch.5,# 9,11,12,23,26,30,32,41,57,and 63.
? Review session,Friday Oct,7 @ 11:00AM in IMS-20
? Chapter 6,Work and Energy
?Forms of energy
?Definition of work
?Work-kinetic energy theorem
?Examples
