Newton's Laws of Motion

M. Rocha   

Physics 1 - Week 2 - Chapters 2, 3, 4 & 5

Aristotle (450 BCE):

Aristotle's cosmology attempted to explain why objects move

Natural Motion - Proceeds  from the nature of the object,  proportional to its weight.

Violent Motion - Imposed motion resulting from pushing or pulling.

All matter is continuous and composed of 4 basic elements (air, water, earth, fire). 

Celestial bodies are made out of quintessence (the fifth essence) and follow different rules. Earth is at rest at the center of the solar system (geocentric model).

Galileo (1564-1642):

Galileo challenged Aristotle's cosmology after being accepted for over 2000 years

  • If there is no interference with a moving object it will keep moving in a straight line forever.
  • Objects fall at the same speed regardless of their weight.
  • Copernicus heliocentric model of the solar system is right. The Earth moves!

Galileo (1564-1642):

Galileo introduced the concept of inertia

  • If there is no interference with a moving object it will keep moving in a straight line forever.

Inertia is the tendency of an object to resists changes in motion 

Inertia

The tendency of an object to resists changes in motion 

  • Is a property of matter  
  • Is proportional to the amount of matter (mass)

Mass is not the same as weight!

Mass and weight are something different

Mass is a property of objects due to how much matter they have

Weight is the gravitational force objects experience due to gravity

Weight

Weight is the gravitational force objects experience due to gravity. Weight is proportional to the mass of objects, but is not the same as mass

Newton (N) is a unit of Force

1 Kg has a weight of 9.8 N on Earth

1 N = 1 \ kg\frac{m}{s^2}

1 Kg of mass is pulled down by the gravity of Earth with a force of 9.8 Newtons.

A more proper definition will come after we see Newton's 2nd law

Mass/Inertia vs. Weight

1. Pull slowly on the string -- what happens?

 

2. Pull quickly on the string -- what happens?

Newton's First Law of Motion

Newton refined Galileo's idea and made it his first law, appropriately called the law of inertia

Every object continues in a state of rest or of uniform speed in a straight line unless acted on by a nonzero net force.

 Net Force & Vectors

Any quantity that requires both magnitude an direction for a complete description is a vector quantity

Force is a vector quantity. When more than a single force acts on an object , we consider the net force

 Adding Vectors 

The sum of two or more vectors is called the resultant. 

To find the resultant of two vectors that don't act in exactly the same or opposite direction , we use the pralelogram rule

-3 N          +

 2 N   

 -1 N   

=>

 2 N   

3 N 

R = \sqrt{X^2 + Y^2} = \sqrt{3^2 + 2^2} = \sqrt{13}

Pythagorean theorem

 2 N   

3 N 

=

Pythagorean Theorem

What is the magnitude of the resultant when adding the two vectors on the screen?

 4000 N   

- 4000 N 

R = \sqrt{X^2 + Y^2} = \sqrt{4000^2 (-1^2 + 1^2)} = 4000\sqrt{2}

Checkpoint

 Vectors vs. Scalars

Vectors - magnitude and direction

Scalars - magnitude

Examples:

Force

Acceleration

Velocity

Examples:

Mass

Volume

Speed

Support Force, A.K.A Normal Force

Atoms on the push back up like a spring!

\sum{F} = 0

When you stand on two bathroom scales, with one foot on each scale and weight evenly distributed, each scale will read

A.  Your weight

B.  Half your weight

C.  Zero

D. Actually more than your weight

Checkpoint

The Equilibrium Rule

When the net force is equal to zero there is not change in motion - a state of rest or constant velocity is maintained 

\sum{F} = F_1 + F_2 + ... = 0

What is the tension force on the right cable?

Checkpoint

T_left + T_right - 500 N - 400 N - 400 N = 0 N

 T_right =  500 N + 400 N + 400 N - T_left = 1300 N - 800 N = 500 N

\sum{F} = F_1 + F_2 + ... = 0
\sum{F} =

The Equilibrium Rule at constant velocity (dynamic equilibrium)

When the net force is equal to zero there is not change in motion - a state of rest or constant velocity is maintained 

\sum{F} = 0
\sum{F} = F_{\mathrm{lift}}+F_{\mathrm{gravity}}+F_{\mathrm{propellers}}+F_{\mathrm{wind}}=0
F_{\mathrm{propellers}}
F_{\mathrm{gravity}}
F_{\mathrm{wind}}
F_{\mathrm{lift}}

When Nellie pushes a crate across a factory floor at constant speed, the force of friction between the crate and the floor is?

A. Less than Nellie's push

B. Equal to Nellie's push

C. Equal and opposite to Nellie's push

D. More than Nellie's push

Checkpoint

Friction

The Equilibrium Rule

When the net force is equal to zero there is not change in motion - a state of rest or constant velocity is maintained 

\sum{F} = 0

Checkpoint

1000

500

500

500

500

1000

Speed, Velocity and Acceleration

 Velocity vs. Speed

Velocity is a Vector - magnitude and direction

Speed is a Scalar - magnitude

\mathrm{Speed} = \frac{\mathrm{distance}}{\mathrm{time}}
\mathrm{Velocity} = \frac{\mathrm{distance}}{\mathrm{time}}

In a specific direction

Instantaneous vs. Average 

\mathrm{Average \ Speed} = \frac{\Delta d}{\Delta t} = \frac{d_\mathrm{final}-d_\mathrm{initial}}{t_\mathrm{final}-t_\mathrm{initial}} = \frac{\mathrm{distance}}{\mathrm{time}}
\mathrm{Instantaneous \ Speed} = \lim_{\Delta \to 0}\frac{\Delta d}{\Delta t}

Average speed is the speed calculated over an interval of  time

Instantaneous speed is when you measure speed over an infinitesimally small interval of time

Checkpoint 1

What was your average speed if you commuted from San Jose to Gavilan (35 miles) in 35 min  

\mathrm{Average \ Speed} = \frac{35 \ \mathrm{miles}}{35 \ \mathrm{minutes}}
= 1 \frac{\mathrm{miles}}{\mathrm{min}}[\frac{60 \ \mathrm{min}}{\mathrm{hour}}] = 60 \frac{\mathrm{miles}}{\mathrm{hour}}

What about your average velocity?

= 60 \frac{\mathrm{miles}}{\mathrm{hour}} \ \mathrm{SE}
\mathrm{Average \ Speed} = \frac{\mathrm{Total \ distance \ covered}}{\mathrm{Time \ interval}}

Acceleration

\mathrm{Acceleration} = \frac{\mathrm{Change \ of \ velocity}}{\mathrm{Time \ interval}} = \frac{\Delta \ v}{\Delta \ t}

Checkpoint 2

If a car goes from rest to 36 km/h in 10s, what is its acceleration? 

\mathrm{Acceleration} = \frac{\Delta \ v}{\Delta \ t} = \frac{ 36 \ \mathrm{km/h}}{10 \ \mathrm{s}} = 3.6 \frac{ \mathrm{km}}{\mathrm{h \cdot s}}[\frac{1 \ \mathrm{h}}{3600 \ \mathrm{s}}]
= 1\times 10^{-3} \frac{ \mathrm{km}}{\mathrm{s}^2}
\mathrm{Acceleration} = \frac{\mathrm{Chance \ of \ velocity}}{\mathrm{Time \ interval}} = \frac{\Delta \ v}{\Delta \ t}

Chekpoint 3

If the Moon travels around the Earth at a constant speed of 1 km/s, is it accelerating? 

Yes! The direction of is motion is changing

Acceleration is a vector!

Newtown's 2nd law

Newtons 2nd Law

Forces cause acceleration

Inetertia resists acceleration

and

\mathrm{Acceleration} = \frac{\mathrm{Net \ Force}}{\mathrm{mass}}
\mathrm{Acceleration} \propto \mathrm{Net \ Force}
\mathrm{Acceleration} \propto \frac{1}{\mathrm{mass}}

Newtons 2nd Law

Acceleration is inversely proportional to mass

\mathrm{Acceleration} \propto \frac{\mathrm{1}}{\mathrm{mass}}
\mathrm{Acceleration} = \frac{\mathrm{Net \ Force}}{\mathrm{mass}}

Free falling objects experience the same acceleration regardless their mass

This is because:

\mathrm{Force \ of \ gravity} \propto \mathrm{mass}
\mathrm{Acceleration} \propto \frac{1}{\mathrm{mass}}

but

So the mass cancels out:

\mathrm{Acceleration} = \frac{g m}{m}

Weight (force due to gravity)

Force = mass x acceleration

Weight = mass x grav. acceleration

Weight = mass x g 

On the surface of Earth

where  g = 9.8 m/s^2  

g is the gravitational acceleration of all objects near the Earth's surface

F_g = mg

Definition of a Newton

1 kg has a weight of 9.8 N on Earth

9.8 \ N = 1 \ kg \times 9.8 \ \frac{m}{s^2}
\mathrm{Force} = \mathrm{mass} \times \mathrm{acceleration}
\Rightarrow 1 N = 1 \ kg \ \frac{m}{s^2}
\mathrm{Force} = \mathrm{weight}= 9.8 N, \ \mathrm{mass} = 1 \ kg, \ a = g = 9.8 \ \frac{m}{s^2}

Newtown's 3rd Law

Forces and Interactions

If you push it pushes you back!

Newtons 3rd Law

To every action there is always an opposed equal reaction

Whenever an object exerts a force on a second object, the second object exerts an equal an opposite force to the first

Or in other words

Newtons 3rd Law: For every action there is a reaction

Force is the same, acceleration is different

acceleration is what causes the damage

Action and reaction of different masses

M

m

a = \frac{F}{m}

Summary of Newton's Laws

1st - Law of Inertia: Every object continues in a state of rest or uniform speed in a straight line unless acted on by a nonzero net force.

2nd - Acceleration is caused by a not zero net force: The acceleration of an object is directly proportional to the net force acting on a object,  is in the direction of the net force, and inversely proportional to the mass of the object.

3rd - For every action there is a reaction: To every action there is always an opposed equal reaction.

a = \frac{F}{m}

My Short Summary of Newton's Laws

For every action there is a reaction: To every action there is always an opposed equal reaction.

a = \frac{F}{m}

Motion under Constant Acceleration

Free Fall

a = g = 9.8 \ \frac{m}{s^2}

Free fall (ignoring air resistance) is an example motion under constant acceleration 

Equations of Motion Under Constant Acceleration

\mathrm{Acceleration} = \frac{\Delta \ v}{\Delta \ t}
\Rightarrow \Delta \ v = \mathrm{Acceleration} \ \Delta \ t
\Rightarrow (v_f \ - \ v_i) = a \ (t_f \ - \ t_i)
\Rightarrow v_f = a \ (t_f \ - \ t_i) + v_i

Velocity aquired

Equations of motion: Velocity acquired

\mathrm{Average \ velocity} = \overline{v} \ = \frac{\Delta \ d}{\Delta \ t}
\Rightarrow \Delta \ d = \overline{v} \ \Delta \ t
\Rightarrow (d_f \ - \ d_i) = \frac{(v_f \ + \ v_i)}{2} \ (t_f \ - \ t_i)
\Rightarrow d_f = \frac{a}{2} \ (t_f \ - \ t_i)^2 + v_i \ (t_f \ - \ t_i) + d_i

Distance traveled

\Rightarrow (d_f \ - \ d_i) = \frac{a (t_f \ - \ t_i) + 2v_i}{2} \ (t_f \ - \ t_i)

Equations of motion: Distance traveled

Free Fall

\Rightarrow v = a \ t

From rest (             )

v_i = 0
d = \frac{a}{2} \ t^2

Set 

t_i = d_i = 0
a = g = 9.8 \ \frac{m}{s^2}
v = a \ t
d = \frac{a}{2} \ t^2

Checkpoint 4

,

t (s) v (m/s) d (m)
0
2
4
6
8
10
0 0
20 20
40 80
60 180
80 320
100 500
a = g = 9.8 \ \frac{m}{s^2} \simeq 10 \ \frac{m}{s^2}
d = \frac{a}{2}t^2 \Rightarrow d \propto t^2

Constant Acceleration

velocity

distance

Time

Time

v = a \ t \Rightarrow v \propto t

Constant Acceleration

velocity (m/s)

Time (s)

v = a \ t \Rightarrow v \propto t

If acceleration is the slope of the velocity vs. time plot how do you find the acceleration  given the plot?

a = \mathrm{slope}
a = \mathrm{slope} = \frac{\mathrm{rise}}{\mathrm{run}} = \frac{100\frac{m}{s}}{10 s} = 10\frac{m}{s^2}

Non-Constant Acceleration

Physics of skydiving

F_{\mathrm{air \ Resistance}} = R
F_{\mathrm{gravity}} = mg
a = \frac{F_\mathrm{net}}{m} = \frac{mg - R}{m} = g-\frac{R}{m}

Is acceleration constant?

R \propto v \Rightarrow a \propto v

What is a when R = 0?

What is a when R/m = g?

What is a when R/m > g ?

No,

a = g, \ \mathrm{freefall}
a = 0, \ \mathrm{terminal \ velocity}
a < 0, \ \mathrm{decelration}

The End

Where does the mass of particles come from?

Inertia

If Galileo and Copernicus are right and the Earth moves, why if you jump next to a wall you don't get slammed by the wall moving at  30 km/s?

Motion is Relative

Are we moving right now?

Relative to what?

Relative to the center of the earth we are moving at 0.33 Km/s

Relative to the sun we are moving at 30 Km/s

Relative to the center of our  Galaxy we are moving at 250 Km/s

Isaac Newton

1642 - 1726

Isaac Newton was an English mathematician, astronomer, and physicist (described in his own day as a "natural philosopher") who is widely recognized as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made pathbreaking contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Vector decomposition

Velocity is a Vector - magnitude and direction

Cross Wind

 Adding Vectors 

If not orthogonal (perpendicular to each other), the parallelogram rule still works but you can not use the Pythagorean theorem to compute the magnitude of the resultant

 2 N   

3 N 

R = \sqrt{X^2 + Y^2}

External Force: ground pushes the apple forward

If action and reaction are equal an opposite, why do they move?