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ASU / Physics / PHY 121 / What is the content of newton´s laws of friction?

# What is the content of newton´s laws of friction? Description

##### Description: Friction, conservation of energy, non-conservative forces, momentum, collisions
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PHY 121 (Mechanical Physics) Exam #2 Study Guide

## What is the content of newton´s laws of friction?

Topics covered in the exam:

1. Newton´s laws with friction

2. Work and energy (non­conservative forces + mechanical energy) 3. Conservation of energy

4. Conservation of momentum and collisions

Theory

Friction  ⃗f  Friction is velocity, surface area independent

∙ Static­ there is no relative motion between the surfaces of contact; friction is static  ( fs )

 Always equal to the amount of force applied

 Typically:  μk<μs

 Model:  fs≤ fsmax=¿ fs≤ μs N

## What is potential energy?

We also discuss several other topics like What is the meaning of p. chytridiomycota?

 μs →  Coefficient of static friction. Directed opposite to motion.

∙ Kinetic­ There is non­zero relative velocity between the surface of contact; then friction is kinetic ( fk )

 Model:  fk=μk N

 μk  = Coefficient of kinetic friction. Constant dependent on materials. Directed  opposite to relative motion.

Example

Work and Energy ∙ Problems based on Newton´s second law:  ⃗F =m ⃗a .

## What is elastic energy according to hook`s law?

Don't forget about the age old question of Why do we care about measurements?

∙ Recall: ⃗a=d ⃗vdt=d2⃗v

d t2

∙ System­ a collection of objects under study whose physics is of interest. ∙ Environment­ is everything else.

∙ Given ⃗F the equation needs to be solved for ⃗r (position) and ⃗v

(velocity)

1. ⃗F  is constant (i.e. weight of an object near Earth,  ⃗w=mg ) 2. ⃗F=⃗F(t) is a function of time (i.e. bat hitting a ball,  ⃗F=F∗t2) 3. ⃗F=⃗F(⃗r) is a function of position (i.e. gravity, electromagnetic, etc.; long range) ∙ Key relationships to keep in mind: Don't forget about the age old question of What is the purpose of maxwell’s equations?
Don't forget about the age old question of Is bayreuth worth visiting?

t

 v=v0+∫ 0

a dt=v0+(Fm )t → x (t )=x0+∫0tvdt=x0+v0t+12 (Fm )t2 (for constant

acceleration)

v

mv dv=12m v2−12m v02 → K=12m v2 If you want to learn more check out What is the meaning of experimental design?

 Kinetic energy (scalar quantity):  ∫ v0

x

 Work:  ∫x 0

F(x )dx=W

 Work­Energy theorem:  W=ΔK

 W=−ΔU  (where U is potential energy)

 2nd Work­Energy theorem:  ΔK =−ΔU → ΔK+ΔU =0 . Thus, total mechanical  energy is defined as  E=K+U . Then,  ΔE=0  when energy is conserved.   Potential energy­ a conservative force where:  F=−dU We also discuss several other topics like What is the meaning of violent motion?

dx   (in 1 dimension).

x f

x fdU

 Also,  W=∫ x 0

F(x )dx=∫ x0

dxdx=−(U( x f)−U ( x0) )=¿W =−ΔU  (system doing

work on its environment)

∙ Potential energy is associated with a system, not an isolated object. PE is a form of  energy that is “retrievable” in the sense that PE stored in a system can be converted into work (KE). PE is the work gravity does:  Ug( y )=mgy.

∙ Elastic energy (Hook´s law)­ The force of an elastic body undergoing a linear  displacement  Δ ⃗x  is opposite to the displacement and proportional to its magnitude  on its environment:  ⃗F=−K Δ ⃗x  where K is a constant

 Stretch:  ⃗F=−K (x−x0)^x

 Compress:  ⃗F=K(x 0−x )^x

 Elastic potential energy:  Ue( x)=12K ( x−x0 )2

W=W c+W nc→−ΔU+W nc=ΔK → ΔK+ΔU=W nc  ; where  ΔK+ΔU=ΔE → ΔE=Wnc

Thus, the energy equation says:

KF +UF=K0+U 0+Wnc

Example

Momentum  ⃗P

∙ The momentum of a particle of mass m with velocity  ⃗v is  ⃗P=m ⃗v ∙ In Newton´s second law language:  ⃗F=d ⃗P

dt

 When m is constant d ⃗P

dt=md ⃗vdt=m ⃗a→ ⃗F=m ⃗a

 Can account when m is not constant (rocket propulsion).

∙ Conservation of momentum: For many particle systems, consider N, masses  mi  and  momenta  ⃗Pi   (i=1,…,N). Let   ⃗Fiext be the external force on the  ith particle, and ∫¿

⃗Fi¿  be the interaction force of  jth on the  ith:

⃗F j i∫¿=d ⃗P

ext=d ⃗Ptot

N⃗Pi →⃗Fnet

dt→ ⃗Ptot=∑i=1 N

dt

⃗Fneti=⃗Fext+∑

¿

j≠ i

⃗Ptot 0=⃗Ptot f

Collisions

∙ One­dimensional elastic collision (key equations):  No external forces =>  ⃗Ptot 0=⃗Ptot f

m1v1i−m2v2i=−m1v1 f +m2v2 f

→ m1( v1i+v1 f)=m2(v2 f +v2i)

 Elastic =>  Ktot 0=Ktot f

2m1v1i2+12m2v2i2 =12m1v1 f

1

m1 ( v1i2 −v1 f

2)=m2(v2 f

2+12m2v2 f 2

2 −v2i2)

∙ One­dimensional inelastic collision:

 Momentum is still conserved

 If objects stick together after the collision it´s a totally inelastic collision  Since KE is not conserved, write ΔK=K f−Ki=Q<0  since ( K f<Ki )  =>  K f=Ki+Q

Example

Summary:

Conservation of energy

Given a force  ⃗F  to a particle of mass m, we have: r f⃗F d ⃗L

∙ Work done by  ⃗F :W =∫

r 0

∙ Kinetic energy of a particle  K=12m v2

∙ If  ⃗F  is conservative, the potential energy U is  F=−dU dx

 Gravity:  Ug=mgy

 Elastic:  Ue=12R Δ x2 (where  R  is a constant)

Work­Energy theorem:

∙ Being  E=W +U  the total mechanical energy and the  ΔE=0  , then KF +UF=K0+U 0

∙ With non­conservative forces KF +UF=K0+U 0+Wnc Momentum  ⃗P

⃗P=m ⃗v

Conservation of momentum:  ⃗Ptot 0=⃗Ptot f

Collisions

 One­dimensional elastic collision

 Momentum:  m1 ( v1i+v1 f)=m2(v2 f +v2i)

Kinetic energy:  m1 ( v1i2 −v1 f

2)=m2(v2 f

2 −v2i2)

 One­dimensional inelastic collision

 Momentum is conserved

 Kinetic energy is not conserved:  K f=Ki+Q  where  Q<0

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