Transcript Document

Summary Lecture 7
7.1-7.6
Work and Kinetic energy
8.2
Potential energy
8.3
Conservative Forces and
Potential energy
8.5
Conservation of Mech. Energy
8.6
Potential-energy curves
8.8
Conservation of Energy
Systems of Particles
9.2
Thursdays 12 – 2 pm
PPP “ExtEnsion”
lecture.
Centre of mass
Room 211 podium
Problems: Chap. 8 5, 8, 22, 29, 36, 71, 51 level
Turn up any time
Chap. 9 1, 6, 82
Outline Lecture 7
Work and Kinetic energy
Work done by a net force results in kinetic energy
Some examples: gravity, spring, friction
Potential energy
Work done by some (conservative) forces can be retrieved.
This leads to the principle that energy is conserved
Conservation of Energy
Potential-energy curves
The dependence of the conservative force on position is
related to the position dependence of the PE
F(x) = -d(U)/dx
Kinetic Energy
Work-Kinetic Energy Theorem
Change in KE work done by all forces
DK  Dw
Work-Kinetic Energy Theorem
Vector sum of all forces
acting on the body
SF
w  x F .dx
xf
i
 xx ma.dx
f
i

x
x i f
dv
m .dx 
dt
x
x i f
dx
m .dv
dt
xi
xf
 m  v .dv  m[1/ 2v ]
2 vf
vi
vf
vi
= 1/2mvf2 – 1/2mvi2
= Kf - Ki
=
DK
Work done by net force
= change in KE
x
Gravitation and work
Work done by me (take down as +ve)
h
F
mg
Lift mass m with
constant velocity
= F.(-h) = -mg(-h)
= mgh
Work done by gravity
= mg.(-h)
= -mgh
________
Total work by ALL forces (DW) =
0
=DK
Work done by ALL forces = change in KE
DW = DK
What happens if I let go?
Compressing a spring
Compress a spring by an amount x
F -kx
x
Work done by me Fdx = kxdx = 1/2kx2
Work done by spring
-kxdx =-1/2kx2
Total work done (DW)
=
0
=DK
What happens if I let go?
Moving a block against friction
at constant velocity
f
F
d
Work done by me
= F.d
Work done by friction = -f.d = -F.d
Total work done
What happens if I let go?
=
0
NOTHING!!
Gravity and spring forces are Conservative
Friction is NOT!!
Conservative Forces
A force is conservative if the work it does on a
particle that moves through a round trip is zero:
otherwise the force is non-conservative
A force is conservative if the work done by it on a
particle that moves between two points is the same
for all paths connecting these points: otherwise the
force is non-conservative.
Conservative Forces
A force is conservative if the work it does on a
particle that moves through a round trip is zero;
otherwise the force is non-conservative
Consider throwing a mass up a height h
-g
h
work done by gravity for round trip:
On way up: work done by gravity = -mgh
On way down: work done by gravity = mgh
Total work done
Sometimes written as
 F.ds  0
= 0
Conservative Forces
A force is conservative if the work done by it on a
particle that moves between two points is the same
for all paths connecting these points: otherwise the
force is non-conservative.
Work done by gravity
Each step
height=Dh
w = -mgDh1+ -mgDh2+-mgDh3+…
= -mg(Dh1+Dh2+Dh3 +……)
-g
h
= -mgh
Same as direct path (-mgh)
Potential Energy
The change in potential energy is equal to minus the
DU = -Dwforce ON the body.
work done BY the conservative
Work done by gravity
h
= mg.(-h)
= -mgh
Therefore change in PE is
mg
Lift mass m with
constant velocity
DU = -Dw
DUgrav = +mgh
Potential Energy
The change in potential energy is equal to minus the
work done BY the conservative force ON the body.
Compress a spring by an amount x
F -kx
x
Work done by spring is Dw = -kx dx = - ½ kx2
Therefore the change in PE is
DU = - Dw
DUspring = + ½ kx2
Potential Energy
The change in potential energy is equal to minus the
work done BY the conservative force ON the body.
DU = -Dw
but recall that
Dw = DK
so that
DU = -DK
or
DU + DK = 0
Any decrease
increase in PE results from a ndecrease
increase in KE
DU + DK = 0
In a system of conservative forces, any
change in Potential energy is compensated
for by an inverse change in Kinetic energy
U+K=E
In a system of conservative forces, the
mechanical energy remains constant
Potential-energy diagrams
Dw = - DU = F. Dx
thus
DU
F 
Dx
In the limit
dU
F 
dx
The force is the negative gradient
of the PE curve
If we know how the PE varies with
position, we can find the conservative
force as a function of position
PE of a spring
dU
F
dx
here U = ½ kx2
Energy
U= ½ kx2
dU
so F  
dx
d 1 2
  (2 kx )
dx
 F  kx
x
Energy
Potential energy
U= ½ kx2
Total mech. energy
At any position x
PE + KE = E
KE
U= ½ kA2
E=
U+K=E
K=E-U
= ½ kA2 – ½ kx2
= ½ k(A2 -x2)
PE
x’
x
x=A
Roller Coaster
K
Fnet=-dU/dt
Et
U
Fnet = mg – R
R = mg - Fnet
K
Et
U
R
Fnet=-dU/dx
mg
Fnet = mg – R
R = mg - Fnet
Conservation of Energy
We said: when conservative forces act on a body
DU + DK = 0 U + K = E (const)
This would mean that a pendulum
would swing for ever.
In the real world this does not happen.
Conservation of Energy
When non-conservative forces are involved, energy can appear
in forms other than PE and KE (e.g. heat from friction)
Energy
converted to
int
other forms
DU + DK + DU
=0
Ki + Ui = Kf + Uf + Uint
Energy may be transformed from one kind to another in
an isolated system, but it cannot be created or destroyed.
The total energy of the system always remains constant.
Stone thrown into air, with air resistance.
How high does it go?
Ei
f mg
=
Ef
+
Eloss
Ki + Ui = Kf + Uf + Eloss
½mvo2 + 0 = 0 + mgh + fh
h
½mvo2 = h(mg + f)
v0
upward
2
0
mv
h
2(mg  f)
Stone thrown into air, with air resistance.
What is the final velocity ?
E’i
=
E’f
+
E’loss
f
K’i + U’i = K’f + U’f + E’loss
0+
mg
mgh =
mv02
2(mg f)
½mvf2 =
½mvf2
+0
= ½mvf2 + f
mv02
mg 2(mg f)
- f
mv02
+ fh

2(mg f)
mv02
2(mg f)
mg
h
mv02
2(mg f)
2
v
0
mg  f 
v f2 
(mg f)
mg  f
v v
mg  f
2
f
2
0
downward
Centre of Mass (1D)
M = m 1 + m2
M
m1
0
x1
m2
x2
xcm
M xcm = m1 x1 + m2 x2
xcm
m1x1  m 2x 2

M
In general
xcm
1
  m i xi
M