Dynamics

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[email protected] Física Aplicada

Dynamics



Objectives

● To understand the concept of force and understand its vectorial character

● To establish the relationship between force and the variables that define the movement

● To understand the meaning of Newton's laws

● To identify the different types of motion in terms of applied forces.

● To know the physical meaning of dynamic quantities

● To know and understand the importance of the principles of conservation of energy, angular momentum, and know how to use for solving problems

● Transfer the concepts used for one point system to a system of material points



Contents

1. Force and Motion: Newton's laws

2. Particle systems

3. Conservation Theorems

4. Work and Energy

BibliographyBibliography

F. Físicos de la Ingeniería II (SPUP-064)F. Físicos de la Ingeniería II (SPUP-064)

Física (Alonso-Finn) Física (Alonso-Finn)

Física (Tipler)Física (Tipler)

Temas Preliminares de Física (CD) (SPUPV-429)Temas Preliminares de Física (CD) (SPUPV-429)



Relationship with other topics

● Mathematics / Physics

● Mechanics– Rigid body dynamics

– Center of mass calculation

Vectorsd r t dt ∫ dr dt

→ Relationship with various scientific and technological areas (Energy / Principle of conservation)



Preliminary concepts

● Mass:● Force:● Momentum:● Moment of a force:● Angular momentum:

m

F⃗

p⃗=mv⃗

L⃗= r⃗× p⃗

F⃗

OP1

M⃗ O( F⃗ )= r⃗×F⃗

r⃗



Newton's Laws

www.schooltube.com/video/2555122616ce8f15ffdb/Force-and-Motion-Newtons-Law-of-Inertia

Δ p⃗=0v⃗=CST

m

v⃗=0

m

First law. Every object

in a state of uniform

motion tends to remain

in that state of motion

unless an external

force is applied to it.

http://www.schooltube.com/video/2555122616ce8f15ffdb/Force-and-Motion-Newtons-Law-of-Inertia



Newton's Laws

m

m

F⃗F⃗

a⃗

a⃗

Second law. The

acceleration of an object is

in the direction of the net

external force acting on it. It

is proportional to the net

external force and is

inversely proportional to the

mass of the object.

F⃗=m a⃗

a⃗=F⃗m



Newton's Laws

Third law. Forces

always occur in equal

and opposite pairs. If

object A exerts a force

on object B, an equal but

opposite force is exerted

by object B on object A.

A⃗=−R⃗

A⃗ −R⃗



Newton's Laws

A⃗=−R⃗

Δ p⃗=0

F⃗=m a⃗

Problems: 1, 4



Third law: The stubborn horse

According to Newton's third law, whatever force I exert on the cart, the cart will exert an equal and opposite force on me, so the net force will be zero and I will have no chance of accelerating the cart



Frictional forces

Fluid: ∣F⃗k∣∝ f (v )

∣F⃗k∣=μk∣N⃗∣

∣F⃗ s∣=∣F⃗app∣<μ s∣N⃗∣

∣F⃗ s , k∣=μ∣N⃗∣Solids:

Static friction:

Kinematic friction:

coefficient of static friction

coefficient of kinetic frictionμk

μs

f⃗

F⃗app



Frictional forces

coefficient of static friction

coefficient of kinetic frictionμk

μs

f⃗

F⃗app

Fluid: ∣F⃗k∣∝ f (v )

∣F⃗k∣=μk∣N⃗∣

∣F⃗ s∣=∣F⃗app∣<μ s∣N⃗∣

∣F⃗ s , k∣=μ∣N⃗∣Solids:

Static friction:

Kinematic friction:



Applications of Newton's Laws

mg

mg sinθ i⃗(− j⃗)mgcosθ

F⃗ s ,maxN

X

Y

N=mgcos θ

F s ,max=m g sinθ

tanθ=μs

Problems: 6, 7



Variable mass system

F⃗=d p⃗dt

● Second law. The acceleration of an object is in the direction of the net external force acting on it. It is proportional to the net external force and is inversely proportional to the mass of the object.

F⃗=m a⃗




v⃗v⃗+ d v⃗

m m+dm

dmv⃗ p

F⃗ F⃗

p⃗ i=m v⃗+dm v⃗ p p⃗ f=(m+dm)( v⃗+d v⃗ )

d p⃗= p⃗ f− p⃗i=md v⃗−( v⃗ p− v⃗ )dm=md v⃗−u⃗ dm

F⃗=d p⃗dt

=md v⃗dt

− u⃗dmdt

the velocity of the impacting material relative to the object

u⃗




Thrust: Rocket propulsion

p⃗i=m v⃗

t t+ d t

v⃗+ d v⃗v⃗

m m−dm

dm

v⃗+ u⃗

The gas is exhausted at a speed u relative to the rocket

v=v0+ u lnm0

m

F⃗=d p⃗dt

=md v⃗dt

− u⃗dmdt

For F⃗=0

Problems: 3,



Conservation of Momentum

If the net external force on a system is zero, the total linear momentum of the system remains constant

R⃗=d p⃗dt

R⃗=0 ⇒ d p⃗=0



Conservation of Angular Momentum

If the net external torque acting on a system is zero, the total angular momentum of the system is constant

d L⃗Adt

=ddt

( r⃗ A× p⃗ )=( d r⃗ Adt × p⃗+ r⃗ A×d p⃗dt )=

=r⃗ A×d p⃗dt

=r⃗ A×F⃗=M⃗ A

M⃗ A=0 ⇒ d L⃗A=0



Systems of particles

Point particles

Extended objects

mi

mi

mj

mk m=∑i=1

n

mi

Forces and momentum

{ R⃗ , M⃗ 0}f⃗ ijf⃗ ji

F⃗ iF⃗ j



The Centre of Mass

r⃗ cm=∑ mi r⃗im

r⃗ cm=∫ r⃗ dmm



G

Centre of gravity

mi g⃗im j g⃗ j

Centre of the parallel sliding vector system

r⃗G=∑i

(migi r⃗ i )

∑i

migi

r⃗ cm=∑i

(mi r⃗ i )

∑i

mi

r⃗ i

r⃗ j

O⃗C=

∑i

(v i*O⃗Pi )

∑i

v i* =

∑i

(v iO⃗Pi)

∑i

v i

Problems: 8, 9, 10



Centre of mass theoremP1O

0

F → External forces

f → Internal forces

f⃗ ijf⃗ ji

F⃗ i

F⃗ jr⃗ j

r⃗ i

F⃗ i+∑j

f⃗ ij=mi a⃗ i

∑i

F⃗ i+∑i , j

f⃗ ij=∑i

mi a⃗ i

R⃗=∑i

mid 2 r⃗idt 2

=

d 2

(∑i mi r⃗ i)dt 2

=d 2

(m r⃗ cm )

dt 2=m

d 2 r⃗ cmdt 2

R⃗=md 2 r⃗ cmdt 2

R⃗=m a⃗cm



Centre of mass theorem

CM

m1m2

m3

mi

...

≡ CM

M

R


R⃗=m a⃗cm

F⃗1 F⃗ 2

F⃗3

F⃗ i

The motion of any object or system of particles can be described in terms of the motion of the centre of mass plus the motion of individual particles in the system relative to the centre of mass



The Centre of MassThe motion of any object or system of particles can be described in terms of the motion of the centre of mass plus the motion of individual particles in the system relative to the centre of mass


R⃗=m a⃗cmhammer



Conservation of Linear Momentum

R⃗=d∑ p⃗idt

=d p⃗cmdt

=d p⃗dt

R⃗=0 → Δ p⃗=0

If the net external force on a system is zero, the total momentum of thesystem remains constant.



Conservation of angular momentum

d L⃗Adt

=ddt

∑i

( r⃗ Ai× p⃗i)=∑i

( d r⃗ Aidt× p⃗ i+ r⃗ Ai×

d p⃗idt )=

=∑i

( r⃗ Ai×d p⃗idt )=∑i

r⃗ i×( F⃗ i+ f⃗ i)=∑i

r⃗ i×F⃗ i=M⃗ A(*)

M⃗ A=d∑i

L⃗Aidt

=d L⃗Adt

M⃗ A=0 → Δ L⃗A=0

If the net external torque acting on a system is zero, the total angular momentum of the system is constant.



Conservation of angular momentum

∑i

r⃗ i× f⃗ i=∑i ( r⃗ i×∑

j

f⃗ ij)=0

r⃗ i× f⃗ ij+ r⃗ j× f⃗ ji=

O

=( r⃗ i− r⃗ j)× f⃗ ij=

= r⃗ ij× f⃗ ij=0

(*)

f⃗ ij

f⃗ jir⃗ ij

r⃗ j

r⃗ i



Conservation theorems

Linear momentum R⃗=d∑ p⃗idt

=d p⃗cmdt

=d p⃗dt

R⃗=0 → Δ p⃗=0

Angular momentum M⃗ A=d∑i

L⃗Aidt

=d L⃗Adt

M⃗ A=0 → Δ L⃗A=0



Conservation theorems: applications



Conservation theorems: applications

p⃗i p⃗f

F⃗L

F⃗

p⃗f

p⃗iΔ p⃗



Work and energy

Work

W AB=∫

A

B

F⃗⋅d r⃗=12m (vB

2−v A

2 )=K B−K A

Power P=dWdt

Kinetic energy: kinetic energy theorem

W AB=∫

A

B

F⃗⋅d r⃗=∫A

B

M⃗⋅d θ⃗

Kinetic energy in a system of particles:

K=∑i

12mi v i

2=

12 (∑i mi)vcm

2+

12∑i

mi v ' i2

Ec≡K

Problem: 13



Potential energy

Potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration.

U=mgh

h

U=12k x2

Problem: 2, 14, 15



Potential energy

F (x)=−dUdx

∫A

B

F (x)dx=−ΔU=U A−UB

Ep≡U

A force is conservative if the total work it does on a particle is zero when the particle moves around any closed path returning to its initial position.



Conservation of mechanical energy

W AB=W T=W c+W nc=Δ K

W T=−Δ U+W nc=Δ K

W nc=Δ K+ Δ U

W nc=0 ⇒ Δ K+ Δ U=0



Conservation of mechanical energy

Potential energy HIGHKinetic energy low

Potential energy lowKinetic energy HIGH

W nc=0 ⇒ Δ K+ Δ U=0

Roller coaster

Problems: 5, 11, 12, 16

Dynamics

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