1

Chapter 10

Newton’s Laws

Class Question

How do units differ from variables?

List 10 clear examples of units and 10 clear examples of variables.

Class Objectives

Learn the relationship between position, velocity, and acceleration

Learn and apply Newton’s First, Second, and Third Laws

Learn some secrets of Calculus

What are the Forces of Nature?

Gravitational Force Electromagnetic Force

– Electrostatic and Magnetic Nuclear Force

– Strong and Weak Friction Spring Tension Push

Newton’s First Law

Law of Inertia

Newton’s 1st Law

Law of Inertia“Bodies remain at rest or in uniform motion in a straight line unless a net force acts on it.”

Thought QuestionsWhy don’t planets move in straight lines?Would they move it straight lines if there were no gravitational force?

Newton’s Cannon

Newton’s Second Law

F = m a

“The amount of acceleration that a force produces depends on the mass of the object being accelerated.”

Newton’s Second Law

Example Questions:– What happens to acceleration if the force is doubled?

Answer:

– ….if the mass is double? Answer:

Newton’s Third Law

Action-Reaction

“Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body.”

Newton’s Third Law - Action/Reaction

Action-Reaction Table

Wall

Gravity

Earth & Moon

Gravity

Universal Gravitational Force

221

r

mmGF

Inverse Square Law

G = 6.67 10-11 N·m2/kg2

Gravity QuestionsGravity Questions

Did the Moon exert a gravitational force on the Apollo astronauts?

What kind of objects can exert a gravitational force on other objects?

The constant G is a rather small number. What kind of objects can exert strong gravitational forces?

Gravity QuestionsGravity Questions

If the distance between two objects in space is doubled, then what happens to the gravitational force between them?

Find the gravitational force between a 90-kg person and the Earth using the inverse square law (Equation 10-37) and Table 10.6. Show your work and include units for your answer.

Definitions

Vector Quantity– a quantity that has both magnitude and direction

Vector– an arrow drawn to scale used to represent a vector

quantity

Scalar Quantity – a quantity that has magnitude but not direction

Vector or Scalar?Vector or Scalar?

Speed……….. Velocity……... Acceleration.. Time…………. Force………… Distance……..

scalar

vector

vector

scalar

scalar

it depends...

Some Definitions

Position - a location usually described by a graphic on a map or by a coordinate system

54321-3 -2 -1

4

3

2

1

00

-1

-2

-3

-4

(-2, -3)

(4, 3)

Some Definitions

Displacement -- change in position,

where 12 rrr

Δ

1r

r

Δ

2r

54321-3 -2 -1

4

3

2

1

00

-1

-2

-3

-4

(-2, -3)

(4, 3)

Example Write an expression for each vector below

and find their magnitudes.

1r

r

Δ

2r

54321-3 -2 -1

4

3

2

1

0

0-1

-2

-3

-4

(-2cm, -3cm)

(4cm, 3cm)

Vector Notation Position Vector

Unit Vectors

Magnitude

kzjyixr ˆˆˆ

k,j,i ˆˆˆ

222 zyxrr

Team Exercise 4.4

Some Definitions

Average velocity

rate of position change with time

scalarTimeElapsed

vectorntDisplaceme

t

rvave

Some Definitions

dt

rd

t

rv

t

0

lim

Instantaneous Velocity

Some Definitions

vspeed

Speed

the magnitude of instantaneous velocity

Example

Suppose that your drive around in a circle in a parking lot at 30mph.

Consider the instant that your car is facing north.

What is your speed? What is your instantaneous velocity ? What is average velocity?

Some Definitions

Average Acceleration - rate of velocity change with time

Instantaneous Acceleration

t

v

tt

vva

12ave

12

dt

vd

t

va

0t

lim

26

Chapter 10

Newton’s Laws

Newton’s Laws

1st – Law of Inertia 2nd – F=ma 3rd – Action Reaction

Kinematic EquationsWhere did they come from?

tavvo0 xxx

tavoo yyy v

2xxo ta

2

1tvxx

oo

2yyo ta

2

1tvyy

oo

Newton’s Laws

1st – Law of Inertia 2nd – F=ma 3rd – Action Reaction

– Demo Metal ball launcher and dropper Maximum Range and the 90-degree-rule Airplane flare:

http://observe.phy.sfasu.edu/courses/phy101/lectures101/Movies/Plane%20Drops%20Flare.wmv

Derivatives

dt

drv

The derivative is associated with the slope of a function.

run

riselope s

dt

dva

Some Derivatives

Powers

Trig Functions

Exponentials

?xn dx

d

?)xsin( dx

d

?ex dx

d

Example

Assume that r(t)=kt where k is a constant.

Plot r versus t. Plot v versus t. Plot a versus t. Write an equation for v(t) and a(t).

Example

Assume that r(t)=Ct3 where C is a constant.

Plot r versus t. Plot v versus t. Plot a versus t. Write an equation for v(t) and a(t).

Example

Assume that r(t)=rosin(t) where ro and are constants.

Plot r versus t. Plot v versus t. Plot a versus t. Write an equation for v(t) and a(t).

Example 10.1

Assume that you...– walk at 3mph for 15 minutes,– then drive at 40mph for 2 hours, and– then ride a bike at 10pm for 45 minutes.

Plot v versus t. Plot r versus t. Do you differentiate or integrate to get r(t)?

Integrals

The integral is associated with the area under the curve.

widthlengtharea

dt vdr dt adv

Some Integrals

Powers

Trig Functions

Exponentials

?dxxn

?dx)xsin(

?dxex

Example

Assume that a(t)=k where k is a constant.

Plot a versus t. Plot v versus t. Plot r versus t. Write an equation for v(t) and r(t).

Newton’s Second Law (A second look)

A non-zero net force will cause a mass to accelerate.

The time-rate-of-change of momentum is proportional to the net force on the object.

amF

dt

pdF

Newton’s 2nd Law for Constant Mass

If mass is constant then

amdt

vdm

dt

)vm(dF

Problem 10.2

A motorcycle moves with an initial velocity of 30m/s.

When its brakes are applied, it decelerates at 5.0m/s2 until it stops.

Plot the position, velocity and acceleration as a function of time.

What is the position, velocity and acceleration 2 seconds after the brakes are applied?

2ooo ta

2

1tvxx tavv o

Multiple Directions

The equations of motion can be written for each direction independently.

Velocity

Position

tavvo0 xxx

tavoo yyy v

2xxo ta

2

1tvxx

oo

2yyo ta

2

1tvyy

oo

Problem 10.3

A girl shoots an arrow upward. It strikes the ground 10.0 seconds later. What was its initial velocity and what was the

maximum height?

2yoyoo ta

2

1tvyy tavv yyoy

Problem 10.4

A bullet is fired vertically into the air and reached a maximum height of 15,000ft.

What was the initial velocity? What assumptions must be made?

2yoyoo ta

2

1tvyy tavv yyoy

Problem 10.5

A man standing on a 200-ft tower throws a ball upward at 40 ft/s.

How long does it take to hit the ground?

Announcements

Homework 12 due on Wednesday.

Remember to get advising.

Project Two is due on December 6th.– That’s next week today.

Project Two

Scoring: – 1 Point for each inch that the projectile travels.

Instructions:– See course home page

Homework 12

http://www.physics.sfasu.edu/astro/OnlineExams/FCI/fci_main.html

Kinematic EquationsWhere did they come from?

tavvo0 xxx

tavoo yyy v

2xxo ta

2

1tvxx

oo

2yyo ta

2

1tvyy

oo

Calculus Basics

Derivative

Indefinite Integral

Definite Integral

1nn nxx dx

d

1n

xx

1nn

1n

a

1n

b

1n

xdxx

1n1nb

a

1nb

a

n

Constant AccelerationEquation of motion is

dt

dva

dvdt a

dtadv

tavv o

The o in v subscript refers to the original or initial value at the beginning of the time interval of interest.

Integrate both sides

where acceleration is constant.

Arranging this equationdt

dxv

dxdtv

dttavdx oo

dttavdx oo

2ooo ta

2

1tvxx

Substituting the velocity equation from the previous page

Integrating both sides

Coil Gun

Final Exam Question– How fast must the projectile be traveling when it

leaves the straw if it is to travel 100 inches horizontally?

Newton’s Laws -- Review

First LawInteria

Second LawF=ma

Third Law– Action/Reaction

Law of Gravity

221

r

mmGF

Team Exercise, 3 min.1. The derivative of velocity with respect to time is:

– position or acceleration

2. By integrating velocity with respect to time we get:– distance traveled or acceleration

3. The derivative of position with respect to time is:– acceleration or velocity

4. Integrating acceleration twice with respect to time is :– velocity squared or distance

5. The derivative is associated with the _________ while the integral is associated with _________

– area under the curve, slope

Homework 12

5. According to Newton’s Second Law, what is the net force on a 2000-lb car if it travels at a constant 60mph for 2 hours?

Homework 12

6. A 1500-kg automobile has a projected frontal area of 1.9 m2 and a drag coefficient of 0.35. It is traveling at 100km/h on a flat road when suddenly both the engine and brakes fail. What is the drag force on the automobile at the moment the brakes fail? The density of air is 1.3 g/liter.

2d AvC

2

1F

Homework 12

Will the drag force increase, decrease or stay the same as time goes by?

What would the drag force be if the medium was water rather than air?

Problem 10.7 (modified)

Assume that a projectile is fired upward at an angle and that air resistance is negligible.

Find y as function of x instead of time using the equations below.

What is the shape of the projectiles path?

2xxo ta

2

1tvxx

oo 2

yyo ta2

1tvyy

oo

Team Exercise (3 minutes)

One dimensional motion– What is the distance traveled in 3 seconds? – What is the acceleration at 1.25 hours?

Sp

eed

, mp

h

3210

010

20

Time, hours

Exercise - Newton’s Laws

A pickup truck is moving with a constant speed of 30 mph along a city street.

You are sitting in the back of the truck and you throw a ball straight upwards at a speed of 20 mph.

Neglecting air resistance:– What will be the path of the ball with respect to the

pickup truck and where will the ball land with respect to the truck? (i.e. what are the x and y values of the peak and final positions)

Solution

Step 1: Draw a Picture

X=0

Y=0

30 mph

20 mph

Solution (cont’d)

The ball follows a parabolic path and remains directly above the truck at all times.

Is a horizontal force acting upon the ball? – Votes for yes?– Votes for no?

Solution

Answer: NO. There is no horizontal force acting on the ball.– The horizontal motion of the ball is the result of its own

inertia. When thrown from the truck, the ball already possessed a horizontal motion, and thus will maintain this state of horizontal motion unless acted upon by a force with a horizontal component (Newton's first law).

Summary: forces do not cause motion (velocity); rather, forces cause accelerations (change in velocity).

Today’s Demos

Inertia– Lead Brick and Hammer– Inertia Bars– Disk and Ring Race– Rotating Platform– Metronome– Balancing A Meter Stick

Rotation Adds Stability– Bicycle Tires– Moving Water– Spin Guns

Center of Mass– State of Texas– Curious George

Next Week– Bed of Nails– Fluid Flasks– And more....

Trebuchet Problem (Modified 10.6)

The projectile is fired at an angle of 35° relative to the ground.

It is fired with a velocity of 100ft/s. Find the following:

– vxo and vyo

– axo and ayo

How long was the projectile in the air? How high did it go? How far did the projectile go?

tavvo0 xxx

tavoo yyy v

2xxo ta

2

1tvxx

oo

2yyo ta

2

1tvyy

oo

Section 10.3 - Forces

Forces are vector quantities.

kFjFiFF zyx

kN5jN10iN10F

What is the magnitude of this force?

Problem 10.1

A car starts from rest and travels northward. It accelerates at a constant rate for 30 seconds

until it reaches a velocity of 55mph. Plot the acceleration, velocity and position as a

function of time.

Next Chapter…

Locate Homework 12

Take 3 minutes to answer questions 1, 2 and 3.

Project Two – The Trebuchet

Final Trebuchet Tweaks

Consider a trough Weight and test your trebuchet Allow for hole and pin adjustments Prepare for rain

The Ideal Trebuchet Problem

In the absence of air resistance, a projectile fired at an angle of 45° will have a maximum range.

What initial speed must a projectile have in order to have a range of 20 yards?

...of 40 yards?

tavvo0 xxx

tavoo yyy v

2xxo ta

2

1tvxx

oo

2yyo ta

2

1tvyy

oo