Summary Lecture 7Summary Lecture 7
7.1-7.67.1-7.6 Work and Kinetic energyWork and Kinetic energy
8.28.2 Potential energyPotential energy
8.38.3 Conservative Forces and Conservative Forces and Potential energyPotential energy
8.58.5 Conservation of Mech. EnergyConservation of Mech. Energy
8.68.6 Potential-energy curvesPotential-energy curves
8.88.8 Conservation of EnergyConservation of Energy
Systems of ParticlesSystems of Particles
9.29.2 Centre of massCentre of mass
7.1-7.67.1-7.6 Work and Kinetic energyWork and Kinetic energy
8.28.2 Potential energyPotential energy
8.38.3 Conservative Forces and Conservative Forces and Potential energyPotential energy
8.58.5 Conservation of Mech. EnergyConservation of Mech. Energy
8.68.6 Potential-energy curvesPotential-energy curves
8.88.8 Conservation of EnergyConservation of Energy
Systems of ParticlesSystems of Particles
9.29.2 Centre of massCentre of mass
Problems: Chap. 8 5, 8, 22, 29, 36, 71, 51
Chap. 9 1, 6, 82
Problems: Chap. 8 5, 8, 22, 29, 36, 71, 51
Chap. 9 1, 6, 82
Thursdays 12 – 2 pm
PPP “Extension” lecture.
Room 211 podium level
Turn up any time
Thursdays 12 – 2 pm
PPP “Extension” lecture.
Room 211 podium level
Turn up any time
Outline Lecture 7Outline Lecture 7
Work and Kinetic energyWork and Kinetic energy
Work done by a net force results in kinetic energy
Some examples: gravity, spring, friction
Potential energyPotential energy
Work done by some (conservative) forces can be retrieved. This leads to the principle that energy is conserved
Conservation of EnergyConservation of Energy
Potential-energy curvesPotential-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
K w
xixf
f
i
f
i
vv
vv vmdvvm ]/[. 221
= 1/2mvf2 – 1/2mvi
2
= Kf - Ki
= KK
Work done by net force
= change in KE
f
i
xx dxFw .
f
i
xx dxma .
f
i
f
i
xx
xx dv
dtdx
mdxdtdv
m ..
Work-Kinetic Energy TheoremF
x
Vec
tor
sum
of
all f
orce
s ac
ting
on
the
body
mg
F
h
Lift mass m with constant velocity
Work done by me (take down as +ve)
= F.(-h) = -mg(-h) = mghWork done by gravity
= mg.(-h) = -mgh ________
Total work by ALL forces (W) = 0
What happens if I let go?
=K
Gravitation and work
Work done by ALL forces = change in KE
W = K
Compressing a spring
Compress a spring by an amount x
Work done by me Fdx = kxdx = 1/2kx2
Work done by spring -kxdx =-1/2kx2
Total work done (W) =
0=K
What happens if I let go?
x
F -kx
Ff
dWork done by me = F.d
Work done by friction = -f.d = -F.d
Total work done = 0What happens if I let go? NOTHING!!
Gravity and spring forces are Conservative
Friction is NOT!!
Moving a block against friction at constant velocity
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
Conservative Forces
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 = 0
Sometimes written as 0ds.F
h-g
Consider throwing a mass up a height h
Conservative Forces
-g
Each step height=h
= -mg(h1+h2+h3 +……)
= -mgh
Same as direct path (-mgh)
Work done by gravity
w = -mgh1+ -mgh2+-mgh3+…
h
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.
U = -w
Lift mass m with constant velocity
Work done by gravity
= mg.(-h) = -mgh
Potential EnergyThe change in potential energy is equal to minus the work done BY the conservative force ON the body.
Therefore change in PE is
U = -w
h
mg
Ugrav = +mgh
Work done by spring is w = -kx dx = - ½ kx2
Potential EnergyThe change in potential energy is equal to minus the work done BY the conservative force ON the body.
Therefore the change in PE is
U = - w
Compress a spring by an amount xF -kx
x
Uspring = + ½ kx2
Potential EnergyThe change in potential energy is equal to minus the work done BY the conservative force ON the body.
U = -wbut recall that
w = K so that
U = -K or
U + K = 0
Any increase in PE results from a decrease in KEdecrease n increase
U + K = 0In a system of conservative forces, any change in Potential energy is compensated for by an inverse change in Kinetic energy
U + K = EIn a system of conservative forces, the mechanical energy remains constant
Potential-energy diagrams
w= - U
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
dx
dUF In the limit
x
UF
thus
= F. x
Energy
x
U= ½ kx2
kxF
dx
dUF
)kx(dx
ddx
dUFso
22
1
PE of a spring
F = -kx (spring force)
here U = ½ kx2
Energy
x
Potential energy
U= ½ kx2
U= ½ kA2
x=A
KE
PE
At any position x
PE + KE = E
U + K = E
K = E - U
= ½ kA2 – ½ kx2
= ½ k(A2 -x2)
x’
Total mech. energy
E= ½ kA2
Et
K
U
Fne
t=-d
U/d
t Fnet = mg – R
R = mg - Fnet
Roller Coaster
Et
K
U
Fne
t=-d
U/d
x Fnet = mg – R
R = mg - Fnet
mg
R
Conservation of Energy
We said: when conservative forces act on a body
U + K = 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)
U + K + Uint = 0
Ki + Ui = Kf + Uf + Uint
Energy converted to other forms
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.
h
f mg
upward
Stone thrown into air, with air resistance. How high does it go?
Ei = Ef + Eloss
Ki + Ui = Kf + Uf + Eloss
½mvo2 + 0 = 0 + mgh + fh
½mvo2 = h(mg + f)
f)2(mgmv
h2
0
v0
h
f
mg
downward
Stone thrown into air, with air resistance. What is the final velocity ?
E’i = E’f + E’loss
K’i + U’i = K’f + U’f + E’loss
0 + mgh = ½mvf2 + 0 + fh
fmgf)(mg
vv
202
f
fmg
fmgvv 2
02f
mg = ½mvf2 + f f)2(mg
mv20
f)2(mg
mv20
½mvf2 = mg - f f)2(mg
mv20
f)2(mg
mv20
f)2(mg
mv20
Centre of Mass (1D)
0x1 x2
xcm
m1M m2
M = m1 + m2
M xcm = m1 x1 + m2 x2
Mxmxm
x 2211cm
In general iicm xmM1
x
Top Related