summer school 2007b. rossetto1 5. kinematics piecewise constant velocity t0t0 tntn titi t i+1 x(t)...

Summer School 2007 B. Rossetto 1

5. Kinematics Piecewise constant velocity

n

n ii=1

Total distance : X = xt0 tnti ti+1

x(t)

x(ti)

ht

xi = v(ti) . h

Distance runned during the time interval

i i+1 i+1 it , t , with t - t = h :

n

n ii=1

X = h v(t )

v(ti) is the slope of the segment.

i(t )v##############


5. Kinematics Instantaneous velocity

i ix i

h 0

x(t h) x(t ) dxv (t ) = lim

h dt

t0 tnti

x(t)

x(t)

h

td

(t) = dtOM

v

############################

.. ..

- definition of velocity

- definition of acceleration

d(t) =

dt

v##############

TLet = x(t), y(t), z(t) OM##############

Tx y zand = v , v , v v

##############

ti+h

v(t)##############

M

O


5. Kinematics Velocity and acceleration

x y z

x y z

2 2 2x y z2 2 2

(t) = x(t) + y(t) + z(t)

dx(t) dy(t) dz(t)(t) = + +

dt dt dt

d x(t) d y(t) d z(t)(t) = + +

dt dt dt

r u u u

v u u u

u u u

####################################### ###

########################################################

####################################### ###

r

r z

(t) = r(t)

dr(t) d (t) dz(t)(t) = + r(t) +

dt dt dt

r u

v u u u

############# #

########################################################

... . . .. ..2

r θ z= r- r θ + 2r θ+ r θ + z(t)γ u u u########################################################

Cartesian

Polar (cf. chap.1 Coordinates, slide 7)


5. Particle motion First law of Newton (inertia principle)

System without int eraction :

v ctt which means

ctt direction

v ctt

v

############################

##############

Define a system (particle, system of particles, solid)

Second law (principle) of NewtonAs a consequence : system with interaction : F

##############

v##############

dm

dt

vF

############################

changes, depending on the inertial mass m:


5. Motion Extension to variable mass systems

System without int eraction : p ctt############################

ddt

p

F

############################

m p v############################

Definition of the momentum of the system:

1st law: principle of conservation of momentum

2nd law: fundamental law of dynamics


5. Kinematics Rotational dynamics

1 - Definition of angular momentum:

L r p##########################################

0 p##############

( and must be evaluated relative to the same point 0)

2 - Fundamental theorem of rotational dynamics:

ddt

L##############

Proof:

L##############

d d d d dm

dt dt dt dt dt

L r p p pp r v v r r r F

######################################################################################################################################################################################

is the torque of the force generating the movement

r

L##############


5. Kinematics Motion under constant acceleration

x

y

0x

yg u############# #

0x 0x(t) v t x

20 0

0y 00x 0x

x x x x1y(t) g v y

2 v v

0y

(parametric equation of a parabola)

0v##############

Double integration and projection:

20y 0

1y(t) g t v t y

2


5. Kinematics Fluid friction

(K / m)tWv(t) (1 e )

K

Wk

(2nd order differential equation with constant coefficients)

Example: free fall of a particle in a viscous fluid.

dvm W K v

dt

t

v(t)

0

Limit speed :

Speed as a function of time :

Wv( )

K

From the second law :

K : shape coefficient (body): viscosity (fluid)

(0)v##############


5. Kinematics Sliding friction

xy

mW g############################

yg u############# #

Frictional force characterized by :

NR##############

TR##############

TR##############

NR##############

T

N

Rtg

R

Example: inclined plane

Static coefficient > dynamic coefficient

Project the fundamental law of dynamics

(2nd Newton law) onto Ox and Oy axes.


5. Kinematics Uniform circular motion

v##############

0

ω##############

ruu

0.

##############

r

rPr oof : derive OM r u##############

22

r rv

r u ur

v ctt v##############

r ctt r

and

(implies ω ctt##############

Acceleration: from chap. I Coordinates, slide #7

Definition of uniform circular motion

rdu

udt

##############

Definition of angular velocity

v uv############################

and

then

dmodulus :

dtω direction : triedre

ω, r, v direct

##############

##########################################

(central)

)

MTheorem: ω v r

##########################################


5. Kinematics Motion under central force (1)

r2m m'

Gr

F u############################

Example: gravitation

.

.ru

O(m)

P(m’)

Theorem slide #5: 2 dctt, r ctt C

dt

L##############

From the 2nd Binet law:2

2 2d u Gm 1

u , with u=rd C

Sketch of proof: expression of acceleration in polar coordinates:2 2.. . . . .. . ..

2 2r 2

du d ur r 2r r and r C , r C u

d d

U U

########################## ##

m: gravitational mass, equal to inertial mass

(3rd Newton law)


5. Kinematics Motion under central force (2)

.F’

(Origin)F

p. cb

2Cwith p= (parameter)

Gm

a

2

0 0C c

and e= (eccentricity, e= ), , initial conditions: =1, =0Gm a

M(r,)

2 2 22 2 2

2 2x y b

(equation of the ellipse: + =1, p= , a =b c , F'A' = a-c)aa b

(ellipse, Origin is one of the focuses F’)A’

0

pr= , r=F'M

1+ e cos( - )

Solution of the differential equation

..

.. .A


5. Work and energy Work

P EF grad##############

AB

W F dl############################

Definition. Work of a force along a curve :

dl##############

F

A

B

F##############

P

HProperty. If there exists EP such that

then is conservative.

AB

Potential energy. We define the potential energy of aconservative force vectorfield as a primitive:

Kinetic energy. The kinetic energy of a particle of mass m and velocity v is defined as Ek=(1/2)mv2.

PE (+ctt) F dl############################

Energy

F##############

summer school 2007b. rossetto1 5. kinematics piecewise constant velocity t0t0 tntn titi t i+1 x(t)...

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