KINETIC ENERGY - Facoltà di Medicina e Chirurgia

Laurea Magistralis
in MEDICINE
and SURGERY
Corso di Laurea
Specialistica
in
MEDICINA
e CHIRURGIA
“HARVEY”
corso integrato
FISICA - disciplina
FISICA
Integrated
Course/Discipline:
PHYSICS
MECHANICS
IInd part
- WORK and ENERGY
- PRINCIPLE of ENERGY CONSERVATION
- KINETIC ENERGY
- CONSERVATIVE FORCES and POTENTIAL ENERGY
- DISSIPATIVE FORCES (FRICTION FORCE)
- MECHANICAL ENERGY CONSERVATION
- MECHANICAL SYSTEM EQUILIBRIUM
D. SCANNICCHIO 2009
01/16
WORK
→
F
→
α
→
s
→
L = F ⋅ s = F s cosα
dimensions [work] = [M][L][t]–2[L] = [M][L]2[t]–2
• I.S. joule (J) = newton x meter
105 x 102
• C.G.S. erg = dyne cm
• PRACTICAL kgmeter = kggravity meter
kggravity = 9.8 N m = 9.8 joule
D.
D. SCANNICCHIO
SCANNICCHIO
2007
2009
02/16
ENERGY
potential capability to develop
mechanical work
ENERGY measure unit same WORK measure unit
ENERGY FORMS:
(revealed directly or in the transformations
from one form to another)
- kinetic
- gravitational potential
- elastic potential
- electric potential
- thermal (heat)
- chemical
- nuclear
- ..........
- ..........
PRINCIPLE of ENERGY CONSERVATION
D. SCANNICCHIO 2009
03/16
ENERGY
PRINCIPLE of
ENERGY CONSERVATION
Etotal = constant
or
∆E = 0
(for an isolated system)
D. SCANNICCHIO 2009
04/16
KINETIC ENERGY
Ek = 1 mv2
2
KINETIC ENERGY THEOREM
(energy conservation)
L = ∆Ek
D. SCANNICCHIO 2009
1
1
2
= Ek2 – Ek1 = mv2 – mv12
2
2
05/16
KINETIC ENERGY
L = ∆Ek = Ek2 – Ek1 = 1 mv22 – 1 mv12
2
2
proof :
rectilinear and uniformly accelerated motion (a = constant)
v1 + v2 ∆t
v2 – v1
∆v
∆s = vmean ∆t =
a=
=
2
∆t
∆t
L = F • ∆s = m a ∆s = m
(v2 – v1) (v1 + v2)
=
2
1
2
2
= m (v2 – v1 ) =
2
2
2
1
1
= m v2 –
m v1 = Ek2 – Ek1 = ∆Ek
2
2
D. SCANNICCHIO 2009
Q.V.D.
06/16
CONSERVATIVE FORCES
z
or
x
y
(1)
A
(3)
LA→ B + LB→ A = 0
(2)
Llinea
closedchiusa
line = 0
→ →
⌠ F⋅
ds = 0
⌡l
l
B
LA → B = f (A,B)
A
B
D. SCANNICCHIO 2009
xA, yA, zA
xB, yB, zB
07/16
DISSIPATIVE FORCES
z
x
y
(1)
or
A
(3)
(2)
Llinea
≠0
closedchiusa
line
B
D. SCANNICCHIO 2009
→
⌠ →F⋅ ds
≠0
⌡l
l
09/16
POTENTIAL ENERGY
LA → B = f (A,B)
A
B
xA, yA, zA
xB, yB, zB
LA → B = f (A) – f (B) ≡ U(A) – U(B) ≡
≡ U(xA, yA, zA) – U(xB, yB, zB)
POTENTIAL ENERGY
U(x , y, z )
D. SCANNICCHIO 2009
08/16
DISSIPATIVE FORCES
EXAMPLE
friction forces
FA = – f v
→
s
→
A
→
B
FA
→
LAB = FA⋅ s = – FA s
Ltotale = – 2 FA s ≠ 0
D. SCANNICCHIO 2009
(closed line)
10/16
GRAVITY POTENTIAL ENERGY
lines
of force
linee
di forza
gravity force
p=mg
z
y
→
x
p
90°
suolo
ground
→
→
L = p ⋅ h = p h = mg h = mg hA– mg hB= U(A) – U(B)
assuming hB = 0 , U(B) = 0
U(A) = mg hA
in general
D. SCANNICCHIO 2009
11/16
GRAVITY POTENTIAL ENERGY
in general:
POTENTIAL ENERGY
due to GRAVITY FORCE
U=mgh
gravity potential energy is linked only to the quote h
above ground (z vertical coordinate), not to the
x and y horizontal coordinates
D. SCANNICCHIO 2009
12/16
MECHANICAL ENERGY CONSERVATION
CONSERVATIVE FORCE FIELD
L = ∆Ek = Ek2 – Ek1
L = U1 – U2
}
Ek1 + U1 = Ek2 + U2
Etotale = U + Ek = constant
in the gravity force field:
D. SCANNICCHIO 2009
13/16
MECHANICAL ENERGY CONSERVATION
Etotale = U + Ek = constant
in the gravity force field:
mgh +
1
2
m v2 = constant
example: body falling
(friction forces neglected)
same for a liquid
falling in a duct
BERNOULLI’s THEOREM
D. SCANNICCHIO 2009
14/16
EQUILIBRIUM CONDITIONS
in a MECHANICAL SYSTEM
translational equilibrium condition
forze conservative
conservative
forces
L = F Δx = U1 – U2 = – ΔU
→
F = – grad U
equilibrio : F = 0
equilibrium:
ΔU = U2 – U1 = 0
D. SCANNICCHIO 2009
F = – ΔU
Δx
ΔU = 0
U1 = U2
15/16
EQUILIBRIUM CONDITIONS
in a MECHANICAL SYSTEM
ΔU = 0
unstable
indifferent
stable
D. SCANNICCHIO 2009
16/16
SMALL OSCILLATIONS around
a STABLE EQUILIBRIUM POSITION
stable equilibrium position :
very small displacement x :
D. SCANNICCHIO 2009
SMALL OSCILLATIONS around
a STABLE EQUILIBRIUM POSITION
dU(x)
F=–
=–kx
dx
F=ma
moto
armonico
harmonic
motion
k
a = – m x = – ω2 x
k
ω= m
x = A sen
sin ωt
•PENDULUM
•ATOMS (in solids and in molecules)
•PARTICLES
(in a medium where elastic waves propagate)
D. SCANNICCHIO 2009
in differential
and: integral form:
inwork
forma
differenziale
→
→
ΔL = F ⋅ Δs
D. SCANNICCHIO 2009
→
→
dL = F ⋅ ds
⌠ → →
L = ⌡ F ⋅ ds
AB