Energy Kinetic (energy of motion) E or KE
1
1
2
2
2
2
K = m v = m(v x + v y + v z )
2
2
example
Units
m2
[kg 2 ] J
s
(Joule)
baseball m=0.15 kg
pitched at
v = 69 mph = 36.5 m/s
1
1
2
K = mv = (0.15)(36.5)2 [kg (m/s)2 ]
2
2
K = 100 J
v = 69 mph
v = 100 mph
K = 210 J
!!! Lethal energies
5/6-1
Work done by force F acting over displacement d
F
OR
d||
F
F||
d
F||
W= F|| d
d
W= F d ||
component (projection of)
F along d
component (projection of)
d along F
W= F d cos()
Units
m
m2
(N) m (Kg 2 ) m Kg 2 J !!
s
s
5/6-2
Collinear F and d
d
F
F
Note kinetic friction force
always opposes displacement
always does - work
d
W= -F d
W= +F d
example
N
box on frictionless plane
T
Constant force not // d
Nd
mg d
note
d
mg
T
T||=T cos()
Wtot=T|| d=T cos() d
5/6-3
Ex: object lowered on a string
T
mg
h
F = T - mg
d = h
Work done by gravity
d
Wg = m g h
(positive work)
Work done by tension
W T = T d = -T h
(negative work)
Total Work Done on Object
W = (T - m g) h
W = Wg - WT
If object lowered with constant velocity,
Net Work
then
m g - T = 0 and W =0
If object accelerates down, then
0 < W < mgd
5/6-4
Work-Energy Theorem
Object: m; constant force F; constant accelertation a=F/m
v - v0
v = v 0 + at a =
t
W = F x = ma x
v + v0
x =[
]t
2
v - v0
v + v0
W =m (
) {[
] t} true for non-const. a also !!!
t
2
m2
1
2
21
W=
(v - v 0 )
W = mv - mv 0 2
2
2
2
1
mv 2 = K = Kinetic Energy
2
W = K
W = K - K0
or
Total work done on object= change in kinetic energy
(work done by total force)
2
2
m
[K] = kg ( 2 ) Joule = Nm
5/6-5
s
ft
[K] = slug ( 2 ) ft lb
s
Power
A measure of the rate at which work is done.
Average Power
W
P=
t
Instantaneous Power
dW
P=
dt
Power= work per unit time [ J/s= Watt]
SI unit: J/s = watt, W
1 horsepower = 1 hp = 746 W
5/6-6
10 N
F
5 kg
m
Example
3m
d
W = F|| d = 10 (3) Nm = 30 J
Assume: no other forces (no friction) & starts from rest (vi = 0)
W= K
What is vf ?
W= F d = mvf2 - mvi2
30 J = (5 kg) vf2
vf2 = 3.5 m/s
If it this took 3 sec what was the average power input by F ?
P = W/ t = 30 J/3s = 10 watts
5/6-7
Object projected up inclined plane with speed v0
How far up ?
Work Energy Theorem
d
v0
Normal force does no work
N//d = 0 : N to d
WN = N|| d = 0
N
Fg = m g (-^)
j
Wg = Fg|| d
- Fg sin
- m g sin
d
Wg = (-m g sin ) d
1
1
2
2
W WN Wg mvf mv 0
2
2
Fg
- 2 m v0 2 = -m g d sin( )
1
Note: h=height up plane
h=d sin( )
V02
d=
2 g sin
d
h
5/6-8
-
1
2
2 m v0
= -m g h
Ball thrown upward near surface of the earth
W = Fd|| mgh
W = Fd|| mgh
mg
mg
y= h
y= h
d
mg
d d||
d||
y=0
y=0
mg
W = Fd|| mgh All these cases have the same the
y= h
y=0
d
d
d||=h and the work done by gravity
is the same = mgh !!!!
d||
In all these cases (by the W -E theorem)
the change in kinetic energy
K=mgh
Indeed, in every case you invent the work done
by gravity will depend only on the vertical
change in height (h) and K=mgh !!!!
!!!!
Gravity is a very dependable
conservative force
Conservative Force in general
3 ways to define a conservative force
F = conservative force if:
1. the work done by F in any round
trip motion of an object is zero
2. the change in kinetic energy, K,
caused by F in any round trip
motion of an object is zero.
3. the work done by F when an
object moves from an initial point
to a final point depends only on
these two points and not on the
path taken between them.
W F = 0
KF = 0
Path
Path
Path 1
conservative force examples:
- Gravity - Electric fields - The force from a spring
Potential Energy & Conservation of Energy
Work Energy
Theorem
Now define
K Wif
change in Potential Energy
K - Wif 0
U - Wif
K U 0
kinetic
energy
change
potential
energy
change
*
*
for conservative forces
*
conserved
E
or
KU E
kinetic potential
energy energy
Energy conservation
Ei Ef
initial & final positions/times
5/6-7
total
mechanical
energy
EKU
Ei conservation
In Ef
Energy the case of gravitational force/potential (near the earth's surface)
U = mg (y-yo)
12
E mv mg (y - yo )
2
kinetic
energy
change
yo is some arbitrary position
where U=0
potential
energy
change
Can choose zero of potential
anywhere one wants
- once choice made keep same
5/6-8
energy conserv260-01-11.ppt
Example
Throw an object up how high does it go?
Ei Ef
1
mv 2 mgh
2
2
vi
h
2g
2
2
y= h
vtop = 0 : know
100
h
2 (9.8)
E f 0 mgh
h 510 m
vi = 100 m/s
U=0
y=0
1
2
E i mvi 0
2
5/6-9
m
s
m
( 2)
s
5/6-10
Energy conservation
fast way to solve for
h vs v
5/6-10a
Example roller coaster starts from rest at top of 1st hill
ho
h1
h3
1
2
Ei = mv o + mg h
2
0
o
1
2
E1 = mv 1+ mg h
2
Ei = mg h o
E i = E1
1 2
mg h o= mv 1 mg h
+
2
1
mv 12 = mg (h -h ) o
1
2
v1 = 2g (h -h )o
1
1
1
2
E3 = mv 3+ mg h 3
1
2
but h 3 = h 0
1
2
E3 = mv 3+ mg h 0
2
E1 = E3
1
2
mg h 0= mv 3+ mg h 0
2
1
0= v 3
0= mv 32
2
Object comes to rest before top of 3rd hill
(when h=h0)
5/6-11
Example: loop the loop
conservation of energy is great
but dont forget Newtons Laws !!!
Q. How high does h have to be to have v go to 0
At top of loop? (wrong question if you want to survive !!)
A. h=2R
But you fall off before you get there !!
5/6-12
loop the loop
at top of loop
5/6-12a
v is min at top of loop
v
N mg
v (increasing)
v (decreasing)
v (max)
Spring: Energy Considerations
Hooks Law
F -kx
x
k2k2
W - x xi
2
2
12
U kx U=0 at x=0, i.e. xi = 0
2
1
12
2
E K U mv kx
2
2
5/6-13
x
Force restores to
equilibrium (x=0)
the larger the
displacement the larger
the restoring force
x
k
k
k2
W F dx - kx dx - x 2 - x 2 x i
2 xi
2
2
xi
xi
U - W
k2k2
x xi
2
2
cyclic transfer :: kinetic energy potential energy
121
2
kA mv max
2
2
v max
k
A
m
Energy
1
2
E mv max
x=0
2
1
E kA 2
x= A
2
x= -A E 1 kA 2
2
1
12
2
E mv kx
2
2
5/6-14
A note on a why a
weight hanging
vertically from a
spring can be treated
like one on a
frictionless surface.
5/6-14a
Now include non-conservative forces in Work Energy Theorem
Work Energy Theorem
W1 + W2++ Wn + Wnc = K
change in KE
conservative
forces
Work done by
non-conservative
force
- U1 - U2-- Un + Wnc = K
non-conservative forces
changes in potential energies
kinetic friction (-work)
associated with conservative forces
rocket, explosion, kick
(+work)
Wnc = K + U1 + U2++ Un
Work done by Wnc = E change in total
non-conservative
mechanical energy
forces
5/6-15
m = 2.5 kG k = 320 N/m
v
non-conservative force example
k=0.25
x= 0
f
xf = 0.075 m
v= ?
Wf=Ef-Ei
(displacement)
[force]
1
NC friction,
- work
1
[- k mg](x f )= kx f 2 - mv 2
2
2
1
1
2
mv kx f 2 k mg x
2
2
320
m
k2
2
(0.075) 2(.25)9.8(0.075)
v
x f 2 k g x f
2.5
s
m
m
v=1.04
s
Wf = - k mg x
5/6-16
non-conservative force example
6-17
non-conservative force example
Consider an elevator that falls from a height h, but
which is brought to rest by the brakes just as it reaches the ground
Wf = E = mgh
free
fall
Fg = mg
hv
if Ff only acts over d then
Ff = frictional braking force
braking
d
Fg = mg
Wf = E = mgh = Ff d
Ff = mgh/d
6-17a
Velocity
before brake
mv2 = mg(h-d)
system
Inclined Plane + a Pulley
N
+
T
T
m2
m1 g cos
m2 g
What is a of objects?
+
m1 g sin
T is an internal force
to plane
N - m1g cos = 0
system F = ma (along the direction of potential motion)
- m2g + m1g sin = (m2 + m1)a
Tot external
force on
system
Tot mass
of system
a=
m1 sin - m2
g m +m
1
2
Ball thrown upward with v0
y= h
dup
Recall
1
21
mg
W = F||d = mvf - mv i 2
2
2
1
- mv 0 2
Wup=-mg h=
W = K
2
ddown
vi
y=0
vf
1
mv f 2
Wdown=+mg h=
2
Wround trip = Wup + Wdown = 0
K round trip = 0
v f = v0
Ball thrown upward with v0
y= h
dup
Recall
1
21
mg
W = F||d = mvf - mv i 2
2
2
1
- mv 0 2
Wup=-mg h=
W = K
2
ddown
vi
y=0
1
mv f 2
Wdown=+mg h=
2
Wround trip = Wup + Wdown = 0
vf
K round trip = 0
mg
v f = v0
W = Fd||
d
d||