Download Dynamics: Why Things Move

Dynamics: Why Things Move
• A force is a push or a pull on an
object.
• Forces cause an object to
accelerate…
• to speed up
• to slow down
• to change direction
• Force is a vector.
• There must be a net force on an
object in order to change the
magnitude or direction of a velocity.
• Net force is the unbalanced (greater
than zero) or resultant force that
acts on an object.
• The unit of force is the Newton (N).
A Newton is equal to a kg  m
s
2
Types of Forces
• Contact forces: involve direct
contact between bodies.
• normal support force; friction
• Field forces: act without
necessity of contact.
• gravity; electromagnetic forces of
attraction and repulsion; strong
and weak nuclear forces
Forces and Equilibrium
• If the net force on a body is zero (add up
all the forces acting on a body and the
resultant force is 0 N), the body is in
equilibrium.
• The acceleration of the body is 0 m/s2;
a = 0 m/s2.
• An object in equilibrium may be moving
relative to us in a straight line at the
same speed (called dynamic equilibrium).
• An object in equilibrium may appear to
be at rest (called static equilibrium).
Balanced Forces
•Balanced forces are equal
and opposite and produce a
net force of zero.
•The acceleration of an object
will be 0 m/s2 if the forces
acting on the object are
balanced
(Fnet = 0 N).
•When balanced forces act on
an object:
the object is either at rest, or
the object is moving with
constant velocity (same speed
in the same direction).
Normal Force on Flat surface
Fn
m·g
Fn = m·g for objects
resting on horizontal
surfaces.
Galileo’s Thought Experiment
Galileo’s Thought Experiment
This thought experiment lead to Newton’s First
Law.
Newton’s First Law: Inertia
• If an isolated object is at rest, it will
remain at rest; if it is in motion, it will
continue moving along a straight line at
constant speed.
• The first law is also called the law of
inertia. Inertia is the property of an
object that resists acceleration (changes
in speed and/or direction).
• The first law applies to an object that has
balanced forces acting on it (the net force
is zero).
Newton’s First Law: Inertia
• For example, the horizontal component of
velocity, which is assumed not to change
during the flight of a projectile, is an
example of Newton’s law of Inertia.
• If ΣF = 0 N, then velocity is constant
(acceleration = 0 m/s2).
• An object with a constant velocity is said
to be in a state of equilibrium; its
direction remains unchanged (it moves
along a straight line); and its speed is
constant.
• Keep in mind that a speed of zero (object
at rest) is a constant speed.
Newton’s First Law: Inertia
•Inertia is why you wear a
seatbelt. Without the
seatbelt, you would
continue to move forward
in the car until an
unbalanced force acts
upon you to slow you
down to rest.
•Without the air bag and
seat belt, when the front
end of the car strikes an
object, the driver would
continue to move forward
until hitting the steering
wheel or the windshield.
•The air bag and seatbelt
help to reduce the
accelerating force on the
driver.
Newton’s First Law: Inertia
•Inertia causes the
pendulum to swing
forward, pushing up
on the locking bar
and causing the bar
to catch a ratchet on
the ratchet wheel.
This locks the seat
belt in place and
prevents the person
from continuing
forward.
The First Law is Counterintuitive
Implications of Newton’s 1st Law
• If there is zero net force on a body,
it cannot accelerate, and therefore
must move at constant velocity,
which means
• it cannot turn,
• it cannot speed up,
• it cannot slow down.
What is Zero Net Force?
The table pushes up
on the book.
A book rests on a table.
FT
Physics
Gravity pulls down
on the book.
FG
Even though there are forces on the book, they are
balanced, therefore, there is no net force on the book.
F = 0
Diagrams
• Draw a force diagram and a free body
diagram for a book sitting on a table.
Force Diagram
N
Free Body Diagram
N
Physics
W
W
Unbalanced Forces
• Unbalanced forces are not equal
and opposite and do produce a
net force.
• The net force is 2 N to the right.
Fnet = 5 N + -3 N = 2 N
• Acceleration is always in the direction
of the net force.
Net Force
Now let’s take a look at what happens
when unbalanced forces do not
become completely balanced (or
cancelled) by other individual forces.
An unbalanced forces exists when the
vertical and horizontal forces do not
cancel each other out.
Example 1
Notice the upward
force of 1200
Newtons (N) is
more than the
downward pull of
gravity (800 N).
The net force is
400 N up.
Example 2
Notice that while the vertical normal force and
gravitation forces are balanced (each are 50 N)
the force of friction results in an unbalanced
force on the horizontal axis. The net force is
20 N left.
Another way to look at balanced and unbalanced forces
Newton’s Second Law: F = m·a
• The vector sum of all the forces acting
on an object or system whose mass is
m produces an acceleration a on the
object.
ΣF
a
m
• Mass can be considered a quantitative
measure of inertia. If the mass is
large the acceleration produced by a
given force will be small.
Newton’s Second Law: F = m·a
• Mass and acceleration are inversely
proportional if the force remains
constant. If the same force is applied to
a large mass and to a small mass, the
large mass will have a small acceleration
and the small mass will have a large
acceleration.
• Ex. F = 10 N; m1 = 2 kg; m2 = 5 kg
a1 = 10 N/2 kg = 5 m/s2 (smaller m, larger a)
a2 = 10 N/5 kg = 2 m/s2 (larger m, smaller a)
Working 2nd Law Problems
1. Draw a force or free body diagram.
2. Set up 2nd Law equations in each
dimension.
Fx = m·ax and/or
Fy = m·ay
3. Identify numerical data.
x-problem and/or y-problem
4. Substitute numbers into equations.
“plug-n-chug”
5. Solve the equations.
Newton’s Third Law
• If object A exerts a force on
object B, then object B exerts an
equal and opposite force on
object A
FA on B = FB on A
• In other words, you cannot touch
without being touched!
• Forces always occur in pairs
acting in opposite directions on
two different objects.
• The hammer exerts a
force on the nail
directed to the right
(Fhn); the nail exerts
an equal force on the
hammer directed to
the left (Fnh).
• The action and reaction
forces NEVER act on
the same object!
• For every interaction,
the forces always occur
in pairs and are equal
and opposite.
• When an automobile accelerates, the force of the tires
on the ground is the action force and the ground below
the tires provides the reaction force.
• The reaction force supplies the force that accelerates the car.
• When you sit on a chair, your weight pushes down on
the chair; the chair pushes up with a force equal to your
weight.
• If the chair pushes up with a force less than your weight, you
would fall through it.
• If the chair pushes up with a force greater than your weight, you
would be pushed up above the seat.
• If you were standing in a small boat and tried to jump
across to the nearby dock, you would fall into the water.
The force you exert on the boat as you jump pushes the
boat away, and the equal and opposite force the boat
exerts on you does not result in any forward motion for
you.
Weight, Mass & Gravity
•The weight of an object is
the downward force the
object experiences on or
near the Earth’s surface
due to the pull of gravity.
•All freely falling bodies,
regardless of their mass,
fall with the same
acceleration due to the pull
of gravity.
Weight, Mass & Gravity
• At a given location, the weight of an
object must be proportional to its
mass.
• Fw (weight) = mass·gravity
• unit = N
• Gravitational acceleration g varies
from one location to another, so the
weight of an object will vary according
to its location. The mass of the object
will remain constant no matter where
the object is located.
Commonly Confused Terms
• Inertia is the resistance of an object
to being accelerated
• Mass is the same thing as inertia (to
a physicist).
• Weight is the gravitational attraction
between masses
inertia = mass  weight
Systems of Connected Bodies
• Newton’s laws also apply to a
group of objects that are
connected or in contact with one
another. You find the
acceleration of the system of
connected objects by dividing the
net force on the system by the
total mass being accelerated.
ΣF
a
m total
Systems of Connected Bodies
• Force acting on m1 has to accelerate m1,
m2, and m3 with acceleration a.
• m1, m2, and m3 will all have the same
acceleration a.
• Ftotal = (m1 + m2 + m3)·a
Systems of Connected Bodies
• Net or resultant force on each mass = m·a
• Ftotal = F1 net + F2 net + F3 net
F1 net  resultant   m1  a
F2 net  resultant   m2  a
F3 net  resultant   m3  a
Systems of Connected Bodies
• Force between m1 and m2 has to
accelerate both m2 and m3 with
acceleration a.
• F1 on 2 = (m2 + m3)·a
Systems of Connected Bodies
• Force between m2 and m3 has to
accelerate m3 only with
acceleration a.
• F2 on 3 = m3·a
Systems of Connected Bodies
• Ftotal + F1 on 2 + F2 on 3 – F2 on 1 – F3 on 2 =
(m1 + m2 + m3)·a
• F1 on 2 equal and opposite to F2 on 1.
• F2 on 3 equal and opposite to F3 on 2.
• Resulting equation:
Ftotal = (m1 + m2 + m3)·a
Systems of Connected Bodies
• F1 > F2, therefore, the acceleration of
all three masses will be to the right in
the direction of the greater force
• Apply Newton’s second law: ΣF = m∙a
F1 – F2 = (m1 + m2 + m3)·a
Systems of Connected Bodies
F1- F2
a=
m1+m2 +m3
• Remember that all three masses will have
the same acceleration.
Systems of Connected Bodies
• To determine F1 on 2, isolate m1 and the
forces that act only on m1 to produce the
acceleration.
• Apply Newton’s second law, with the
direction of the acceleration as positive:
ΣF = m∙a;
F1 – F2 on 1 = m1·a
F2 on 1 = F1 - m1·a
Direction of F2 on 1 is to the left
Systems of Connected Bodies
• Newton’s third law
explains that F1 on 2
is equal in magnitude,
but opposite in
direction, to F2 on 1.
• To determine F1 on 2, isolate m2 and the
forces that act only on m2 to produce the
acceleration.
• Apply Newton’s second law, with the
direction of the acceleration as positive:
ΣF = m∙a;
F1 on 2 – F3 on 2 = m2·a
F3 on 2 = F1 on 2 – m2·a
Direction of F3 on 2 is to the left
Systems of Connected Bodies
• Newton’s third law
explains that F2 on 3
is equal in magnitude,
but opposite in
direction, to F3 on 2.
• To determine F2 on 3, isolate m3 and the
forces that act only on m3 to produce the
acceleration.
• Apply Newton’s second law, with the
direction of the acceleration as positive:
ΣF = m∙a;
F2 on 3 – F2 = m3·a
F2 on 3 = F2 – m3·a
Direction of F2 on 3 is to the right
• The procedure for determining the forces
acting between the masses works for any
type of contact forces found between
masses.
Example: Comparing Tensions
Blocks A and B are connected by
massless String 2 and pulled
across a frictionless surface by
massless String 1. The mass of
B is larger than the mass of A.
Is the tension in String 2
smaller, equal, or larger than
the tension in String 1?
The blocks must be accelerating to
the right, because there is a net
force in that direction. We use
the massless string approximation
to directly relate the string
tensions on A and B due to
String 2:
TA on B=TB on A
(FA net)x=T1-TB on A
= T1-T2 = mAaAx
so T1 = T2 + mAaAx
Therefore, T1 > T2.
Free-Body Diagrams
Drawing a Free-Body Diagram
 Identify all forces acting on the object.
 Draw a coordinate system. Use the axes defined in
your pictorial representation. If those axes are tilted, for
motion along an incline, then the axes of the free-body
diagram should be similarly tilted.
 Represent the object as a dot at the origin of the
coordinate axes. This is the particle model.
 Draw vectors representing each of the identified
forces. Be sure to label each force vector.
 Draw and label the net force vector Fnet. Draw this
vector beside the diagram, not on the particle. Or, if
appropriate, write Fnet = 0.
 Then check that Fnet points in the same direction as
a
the acceleration vector
on your motion diagram.
Identifying Forces
 Identify “the system” and “the environment.” The system is the
object whose motion you wish to study; the environment is
everything else.
 Draw a picture of the situation. Show the object—the system—
and everything in the environment that touches the system. Ropes,
springs, and surfaces are all parts of the environment.
 Draw a closed curve around the system. Only the object is inside
the curve; everything else is outside.
 Locate every point on the boundary of this curve where the
environment touches the system. These are the points where the
environment exerts contact forces on the object.
 Name and label each contact force acting on the object. There is
at least one force at each point of contact; there may be more than
one. When necessary, use subscripts to distinguish forces of the
same type.
 Name and label each long-range force acting on the object. For
now, the only long-range force is weight.
The Forces on a Bungee Jumper
T
w
The Forces on a Skier
n
T
fk
w
The Forces on a Rocket
Fthrust
D w
Using Free-Body Diagrams
F
 Fnx  Fgx  Fx  ma x
x
0  0  Fx  ma x
Fx
ax 
m
F
y
 Fny  Fgy  Fy  ma y
Fn  Fg  0  ma y  0
Fn  Fg
The vector sum of the
forces of a free-body
diagram is equal to ma.
Example: A Dogsled Race
During your winter break, you enter a dogsled race
in which students replace the dogs. Wearing cleats for
traction, you begin the race by pulling on a rope
attached to the sled with a force of 150 N at 25° above
the horizontal. The mass of the sled-passenger-rope
system is 80 kg, and there is negligible friction between
the sled runners and the ice.
(a) Find the acceleration of the sled
(b) Find the magnitude of the normal force exerted
on the surface by the sled.
Fn  Fg  F  ma
Fx  F cos   max Fy  Fn  mg  F sin   ma y  0
ax  F cos  / m  (150 N)(cos 25) /(80 kg)  1.7 m/s 2
Fn  mg  F sin   (80 kg)(9.81 m/s 2 )  (150 N)(sin 25)  720 N
Example: Unloading A Truck
You are working for a big delivery company,
and you must unload a large, fragile package
from your truck, using a 1 m high frictionless
delivery ramp. If the downward component of
velocity of the package is greater than 2.50 m/s
when it reaches the bottom of the ramp, the
package will break.
What is the greatest angle q at which you
can safely unload?
Fnx  Fgx  0  mg sin   max
ax  g sin 
x  h / sin 
vx  2ax x  2( g sin  )(h / sin  )  2 gh
vd  vx sin 
 max  arcsin[vd / vx ]
 arcsin (2.5 m) / 2(9.81 m/s 2 )(1.0 m) 


 34.4
Holt Visual Concepts
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Force
Contact and Field Forces
Free Body Diagrams
Newton’s First Law
Mass and Inertia
Newton’s Second Law
Action and Reaction Forces
Newton’s Third Law
Comparing Mass and Weight
Normal Force
• The Physics Classroom: Newton’s
First Law Animations
• The Car and the Wall
• The Truck and Ladder
Drag Force & Terminal Speed
• Drag forces slow an object down as it passes
through a fluid as the fluid exert a kind of
friction as the object moves through them.
• This “friction” is called drag and exerts a drag
force on the object.
• Drag forces:
• act in opposite direction to velocity.
• are functions of velocity.
• impose terminal velocity.
• We will only look at the simplistic case of a
spherical object moving relatively quickly
through air.
Drag Force & Terminal Speed
• In this type of situation, the formula for the
drag force is:
1
2
FD   C    A  v
2
where ρ is the density of air, A is the effective
cross-sectional area of the body and v is the
velocity (really the speed) of the body.
• C is called the drag coefficient and is not truly
a constant (it varies in a complex manner as a
function of the body’s shape, speed and
density).
Drag Force & Terminal Speed
• The drag force will act
against the force that is
accelerating an object
through a fluid.
• If we consider the case of
free-fall, then we can see
that the drag force acts
against the force of gravity
that is accelerating an
object towards the Earth.
• The figure shows the forces
acting on a falling object.
No animals were harmed in
the making of this lecture…
Drag Force & Terminal Speed
• At some point, the
speed of the falling
object becomes
large enough that
the drag force
equals the force
from gravity.
• At that point the
object will no
longer accelerate
but rather
continues to fall at
a constant speed.
Drag Force in Free Fall
When the drag force and the force from gravity
become equal, the falling object has reached what is
called its terminal speed vt
speed is given
FD This
by:
FD
FD
m·
g
m·g
vt 
m·g m·g
2  Fw
C A
Drag as a Function of Velocity
• FD = b·v + c·v2
• b and c depend upon:
• shape and size of object.
• properties of fluid.
• b is important at low velocity.
• c is important at high velocity.
Drag Force in Free Fall
FD = b·v + c·v2
FD
for fast moving objects
for slow moving objects
FD = c·v2
FD = b·v
c = ½·D··A
where
D = drag coefficient
 = density of fluid
A = cross-sectional area
mg