Chapter 3Newton’s Laws of Motion

Why things move the way the do.

Newton’s First Law of Motion

“Maintaining the status quo”

Newton’s First Law of Motion“The Law of Inertia”

“An object at rest will remain at rest and an object in motion will continue to move at a constant velocity, unless acted upon by a net external force.”

The property of an object that causes it to resist changes in its motion is called inertia.

Inertia is proportional to an object’s mass.

Inertia causes motion at a constant velocity.

Fnet =0→ a =0→ constant velocity

Newton’s Second Law of Motion

“How is Acceleration related to Force?”

Newton’s Second Law of MotionThe law of accelerated motion

“The acceleration of an object is directly proportional to and in the same direction as the net force acting on it and inversely proportional to the object’s mass.”

r a =

r Fnetm

r F net =m⋅

r a

Units of Force :

kg⋅ms2

=kg⋅ms2

=Newton,N

Force is a vector and uses the same sign convention as velocity and acceleration.

⇒ effect on

r v

parallel magnitude increases

opposite magnitude decreases

⎧ ⎨ ⎩

direction of r F net

⇒ direction of r a

r F net =

r F1 +

r F2 +

r F3L

r F 1 ,

r F 2 ,

r F 3 are + or −

.based on their directions

SummaryConstant Velocity

Variables:

d

v

t

Relationships:

v =dt

d =v⋅t

t =dv

Constant Acceleration

Variables: Relationships:

v =dt

v =vi + vf

2d = v ⋅t t = d

v a =

vf −vit

vf =vi + a⋅td =vi ⋅t+

12 a⋅t2

v f2 =vi

2 + 2ad

d

v

v i

v f

a

t

Caused by Inertia

Fnet

m r F net

m r F net =0

r a =0

Caused by Net Force

r F net =

r Fi∑

a = r Fnetm

constant

Describing Forces

All forces between objects can be placed into two broad categories:

Contact forcesAction-at-a-distance forces

Contact forces are types of forces in which the two interacting objects are physically contacting each other.

Frictional ForceTensional ForceAir Resistance ForceSpring ForceBuoyant Force

Action-at-a-distance forces are types of forces in which the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite a physical separation.

Gravitational ForceElectric ForceMagnetic Force

Types of ForcesWeight

Friction

Fluid Resistance

Buoyant Force

Mass and WeightMass is a measure of the amount of matter in an object and is related to the object’s inertia.

Weight is a measure of the gravitational force acting on an object.

F=m⋅a↓w

↓m= g

↓⋅

The weight of an object equals the product of its mass and the acceleration due to gravity.

force of gravityweight, w

acceleration due to gravity, g

FrictionThe force that opposes.

Friction is a force whose direction is ALWAYS opposite to the direction an object is moving or would tend to move if there were no friction.

Friction is force between two surfaces:1. the surface of the object and2. the surface on which it is moving

Friction depends on the characteristics of the two surfaces and the force pressing them together.

Static friction exists between a stationary object and the surface on which it rests.

Friction depends on whether the object is moving or stationary.

Static friction must be overcome before an object can begin moving.

Kinetic friction exists between a moving object and the surface on which it is moving.

Kinetic friction is always opposite to the object’s velocity.

Kinetic friction is usually less than static friction.

Static friction is always opposite to the direction the object would move if there was no friction.

Static friction equals the net applied force up to its maximum value which depends on the mass of the object and the properties of the two surfaces.

fkinetic =μkinetic⋅m⋅g

fstatic(max) = μstatic⋅m⋅g

μ static > μkinetic

Fluid ResistanceFluid (liquid or gas) resistance is a frictional force that an object experiences as it moves through a fluid (e.g. air or water).

Fluid resistance depends on the size and shape of the object, density of the fluid, and is directly proportional to the object’s velocity and in the opposite direction.

Example

An object falling through air experiences an upward force of air resistance in addition to the downward force of gravity.

gravitational force

air resistance

r f fluid ∝−

r v

Initially air resistance is small because the velocity is small.air resistance ∝velocity

As the magnitude of the downward velocity increases the force of air resistance increases.

gravitational force > air resistance

net force = gravitational force - air resistanceObject accelerates downward magnitude of

velocity increases

If the object falls far enough the force of air resistance becomes equal to the gravitational force.

At this point:

gravitational force = air resistance

net force, Fnet = 0

a = 0

velocity is constant

This constant velocity reached by an object falling through a fluid is called its terminal velocity.

Buoyant Force

An object immersed in a fluid (liquid or gas) experiences an upward buoyant force.

The buoyant force depends on the volume of the object immersed in the fluid (volume of fluid displaced) and the density of the fluid.

Consider an object dropped into a tank containing a fluid (e.g. oil)

Buoyant Force Only

Buoyant Force & Fluid Resistance

buoyant forcer F B =ρfluid⋅g⋅Vdisplaced

Buoyant Force with Fluid Resistance

The mass of the object is 10kg and the buoyant force is 190N.

If the object starts from rest at a height of 63.5m above the surface of the oil how long will it take for it to reach a depth of 60m in the oil? (Ignore any buoyant force due to air and any fluid resistance.)

Part One of the Motion

Falling from rest a distance of 63.5m to the surface of the oil.

The only force acting on the object is the gravitational force…this part of the motion is Free-Fall.

Part Two of the Motion

Falling a distance of 60m from the surface of the oil.

There are two forces acting on the object:•The gravitational force = mg•The buoyant force = 190N

The object’s velocity as it enters the oil = -35.3m/s

Variables: Relationships:

v =dt v =

vi + vf2

d = v ⋅t t = dv

a =vf −vi

tvf =vi + a⋅t

d =vi ⋅t+12 a⋅t2 v f

2 =vi2 + 2ad

d

v

v i

v f

a

t

m

Fnet

r F net =

r Fi∑

a = r Fnetm

=−63.5m

=0

=10kgFind t=3.6s

find vf=−35.3 m

s

=−98N

=−9.8 ms2

=FG = w = mg

=(10kg) × (−9.8 ms2 )

=−98N

=−98N10kg

=−9.8 ms2

d =vi ⋅t+12 a⋅t2

Substituting values w/o units.

−63.5 = 0 ⋅t + 12 (−9.8 m

s2 )t2

−63.5 = −4.9t2

t2 =−63.5−4.9

t = −63.5−4.9 =3.6s

vf =vi + a⋅t

vf =0 +(−9.8 ms2

)⋅(3.6s) =−35.3 ms

Variables: Relationships:

v =dt v =

vi + vf2

d = v ⋅t t = dv

a =vf −vi

tvf =vi + a⋅t

d =vi ⋅t+12 a⋅t2 v f

2 =vi2 + 2ad

d

v

v i

v f

a

t

m

Fnet

r F net =

r Fi∑

a = r Fnetm

=10kgFind t

=−60m

=−35.3 ms

=FG + FB

=−98N + 190N=92N

= 92N10kg

=9.2 ms2

=92N

=9.2 ms2

d =vi ⋅t+12 a⋅t2

Substituting values w/o units.

−60 = −35.3 ⋅t + 12 (9.2)t2

−60 = −35.3 ⋅t + 4.6 ⋅t2

Writing in the General Quadratic Form

4.6⋅t2 −35.3⋅t+60 =0

Comparing to ax2 +bx+ c=0 gives:a = 4.6

b = -35.3

c = 60

Substituting into the Quadratic Formula

x = -b± b2 −4⋅a⋅c2a gives:

t =-(-35.3)± (−35.3)2 −4⋅(4.6)⋅(60)

2(4.6)

Simplifying gives:

t = 35.3±11.929.2

t1 = 35.3 +11.929.2 =5.1s

t2 = 35.3−11.929.2

=2.6s

The time for the object to reach the surface of the oil was 3.6s.

Therefore the object is at a depth of 60m in the oil at two times.

t1 =3.6s+ 5.1s=8.7s

t2 =3.6s+ 2.6s=6.2s

What is the significance of the two times?

What is the maximum depth the object will reachAnd when does it reach this maximum depth?

Variables: Relationships:

v =dt v =

vi + vf2

d = v ⋅t t = dv

a =vf −vi

tvf =vi + a⋅t

d =vi ⋅t+12 a⋅t2 v f

2 =vi2 + 2ad

d

v

v i

v f

a

t

m

Fnet

r F net =

r Fi∑

a = r Fnetm

=10kg

=−35.3 ms

=92N

=9.2 ms2

=0

Find “d”=−67.7m

Find “t”

t =vf −vi

a

v f2 =vi

2 + 2ad

d =vf2 −vi

2

2a =0 − (−35.3 m

s )2

2 ⋅(9.2 ms2 )

=−67.7m

t =vf −vi

a =0 − (−35.3 m

s )

9.2 ms2

=3.84s

The object reaches the surface in 3.6s.

The object is at a depth of 60m moving downward in 6.2s

The object reaches its maximum depth of 67.7m in 7.44s.

Time to reach maximum depth:

3.84s+ 3.6s=7.44s

The object is at a depth of 60m moving upward in 8.7s.

Newton’s Third Law of Motion

“Forces always come in pairs.”

Newton’s Third Law of Motion“Action and Reaction”

“If object A exerts a force on object B, object B must exert an equal and opposite force on object A.”

“For every action there is an equal and opposite reaction.”

Why do the two forces (action and reaction) not cancel each other?

They act on DIFFERENT objects!

r F A→ B =−

r FB→ A

Newton’s Third Law says that action and reaction forces are always equal and opposite.

It does NOT say that their effects are equal!

Consider two objects moving along a horizontal track.

FA→ B ⇒ aB depends on mB

aB =FA→ BmB

FB→ A ⇒ aA depends on mA

aA =FB→ AmA