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# Chapter 10 Newton's Laws

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```									The Trebuchet
Engineering 111

   Bring your laptops to class on Wednesday.
   Complete Homework 8 before Wednesday.
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
Chapter 10

Newton’s Laws

5
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 Questions
Why 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

   SF = 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?

–   ….if the mass is double?
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

Which exerts a stronger force on the other?
 Earth or Moon
 Mac Truck or VW Bug during Collision
 Car and a Mosquito during Collision
Newton’s Laws

   1st – Law of Inertia
   2nd – SF=ma
   3rd – Action Reaction
Gravity

   Universal Gravitational Force
m1m 2
FG 2
r
Inverse Square Law

G = 6.67  10-11 N·m2/kg2
Gravity 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 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
Chapter 10

Newton’s Laws

18
Newton’s Laws

   1st – Law of Inertia
   2nd – SF=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
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?

   Speed………..       scalar
   Velocity……...    vector
   Acceleration..   vector
   Time………….        scalar
   Force…………        scalar
   Distance……..     it depends...
Some Definitions

   Position - a location usually described by a
graphic on a map or by a coordinate system
4
3
(4, 3)
2

1
0
0
-3     -2       -1   -1       1   2   3     4      5

(-2, -3)        -2
-3
-4
Some Definitions

   Displacement -- change in position,
       
where Δ r  r2  r1
4
3

2                        (4, 3)
r2
1              
0
Δr
              0
-3     -2        -1   -1       1        2   3     4      5
r1
(-2, -3)         -2
-3
-4
Example
    Write an expression for each vector below
and find their magnitudes.
4

3
(4cm, 3cm)
2           
r2
1             
0            Δr
0
-3     -2          -1   -1       1        2   3     4          5
r1
-2
(-2cm, -3cm)
-3

-4
Vector Notation
   Position Vector
 ˆ ˆ ˆ
r  x i  yj  zk
   Unit Vectors

i jˆ
ˆ, ˆ,k
   Magnitude

r  r  x y z
2  2   2

Team Exercise 4.4
Some Definitions

Average velocity
rate of position change with time


         r   Displaceme nt vector 
v ave       
t   Elapsed Time scalar 
Some Definitions

Instantaneous Velocity

    
         r   dr
v  lim      
t 0 t   dt
Some Definitions

Speed
the magnitude of instantaneous velocity


speed  v
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 instantaneous velocity ?
   What is average velocity?
Some Definitions

Average Acceleration - rate of velocity change
with time
          
      v 2  v1   v
aave           
t 2  t1   t

Instantaneous Acceleration
    
         v   dv
a  lim      
t 0 t   dt
Kinematic Equations
Where did they come from?

v x  v x0  a xo t
v y  v yo  a yo t
1
x  x o  v xo    t  a xo t 2

2
1
y  y o  v yo    t  a yo t 2

2
Chapter 10

Newton’s Laws

32
Derivatives

The derivative is associated with the slope
of a function.

dr               dv
v               a 
dt               dt
rise
slope 
run
Some Derivatives

   Powers           d n
x ?
dx
   Trig Functions   d
sin( x )  ?
dx
   Exponentials     d x
e ?
dx
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(wt)
where ro and w are constants.
   Plot r versus t.
   Plot v versus t.
   Plot a versus t.
   Write an equation for v(t) and a(t).
Project Two - Week Two
   Tuesday          Wednesday
–   11:00AM       –   11:00AM
–   11:30AM       –   11:30AM
–   12:00PM       –   12:00PM
–   12:30PM       –   12:30PM
–   1:00PM        –   1:00PM N/A
–   1:30PM        –   1:30PM N/A
–   2:00PM        –   2:00PM N/A
–   2:30PM        –   2:30PM
–   3:00PM        –   3:00PM
–   3:30PM        –   3:30PM
–   4:00PM        –   4:00PM
–   4:30PM        –   4:30PM
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.

 dr      v dt          dv       a dt

area  length  width
Some Integrals

   Powers
 x dx  ?
n

   Trig Functions
 sin(x)dx  ?
 e dx  ?
   Exponentials       x
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).
Team Exercise

43
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?
1
v  vo  a t     x  x o  v o t  ao t 2
2
Multiple Directions

The equations of motion can be written for each
direction independently.
Velocity v x  v x  a x t
0      o

v y  v yo  a yo t

Position                         1
x  x o  v xo t       a xo t 2
2
1
y  y o  v yo t      a yo t 2
2
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?

1
v y  v yo  a y t   y  y o  v yo t  a yo t 2
2
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?

1
v y  v yo  a y t   y  y o  v yo t  a yo t 2
2
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.

   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:
Homework 12

   http://www.physics.sfasu.edu/astro/OnlineExa
ms/FCI/fci_main.html
Kinematic Equations
Where did they come from?

v x  v x0  a xo t
v y  v yo  a yo t
1
x  x o  v xo    t  a xo t 2

2
1
y  y o  v yo    t  a yo t 2

2
Calculus Basics

   Derivative
d n
x  nx n 1
dx
   Indefinite Integral
x n 1
 x  n 1
n

   Definite Integral
n 1 b
b n 1 a n 1
b
x

a
x n dx             
n 1 a n 1 n 1

Constant Acceleration
Equation of motion is
dv
a 
dt
where acceleration is constant.
a dt  dv
Integrate both sides
 dv   a dt
v  vo  a t

The o in v subscript refers to the original or initial value at
the beginning of the time interval of interest.
dx
v 
dt
Arranging this equation
v dt  dx

Substituting the velocity equation from the previous page
dx  v o  ao t  dt
Integrating both sides

 dx   v   o    ao t  dt

1
x  x o  v o t  ao t 2
2
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 Law                Law of Gravity
Interia
m1m 2
   Second Law                  FG 2
F=ma                        r
   Third Law
–   Action/Reaction
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.
1
F  C d Av 2

2
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?

1                             1
x  x o  v xo t  a xo t 2   y  y o  v yo t  a yo t 2
2                             2
Team Exercise (3 minutes)

   One dimensional motion
–   What is the distance traveled in 3 seconds?
–   What is the acceleration at 1.25 hours?
20

0    1     2      3

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

20 mph

30 mph
X=0

Y=0
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?
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                          Center of Mass
–   Lead Brick and Hammer         –   State of Texas
–   Inertia Bars                  –   Curious George
–   Disk and Ring Race           Next Week
–   Rotating Platform             –   Bed of Nails
–   Balancing A Meter Stick       –   And more....
–   Bicycle Tires
–   Moving Water
–   Spin Guns
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.   v x  v x  ax t
0      o

   Find the following:                       vy  v y  ay t
–   vxo and   vyo                                  o      o

–   axo and    ayo
   How long was the projectile in the air?
   How high did it go?
1
   How far did the projectile go?   x  x o  v xo t  a xo t 2
2
1
y  y o  v yo t  a yo t 2
2
Section 10.3 - Forces

   Forces are vector quantities.

ˆ
ˆF ˆF k
F  Fx i y j z
   What is the magnitude of this force?

F  10 N ˆ   10 N ˆj  5N k
i                      ˆ
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.
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?
1
v x  v x0  a xo t    x  x o  v xo t    a xo t 2
2
v y  v yo  a yo t                      1
y  y o  v yo t  a yo t 2
2
Question 1 – Drop Time

   A – ½ time for heavy ball
   B – ½ for lighter ball
   C – same for both balls
   D – considerably less time for heavy ball
   E – considerably less time for lighter ball
Question 2 – Roll off landing position

   A – same horizontal distance for both
   B – heavier ball hits half horizontal distance
   C – lighter ball hits half horizontal distance
   D - heavier ball hits considerably closer to base
   E - lighter ball hits considerably closer to base
Question 3 – Stone Drop Speed
   A – reaches maximum speed soon
   B – speeds up because gravity gets
considerably stronger closer to the Earth
   C – speeds up because of an almost constant
force of gravity
   D – falls because of the natural tendency of all
objects to rest on the surface of the earth
   E - falls because of the combined effects of the
force of gravity and the force of the air pushing
it downward.
Question 4 – Truck and Car Collide

   A – truck exerts more force on car
   B – car exerts more force on truck
   C – neither exerts a force on the other
   D - the truck exerts a force on the car but the
car does not exert a force on the truck.
   E - the truck exerts the same amount of force
on the car as the car exerts on the truck

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