Dinamika

DynamicsDynamics
z After studying Kinematics, we know how to describe
motion in two and three dimensions.
z But what causes this motion?
z Dynamics is the study of the relationship between
motion of objects and the cause of the motion
(forces).
z We will use kinematics' quantities such as displacement,
velocity, and acceleration.
z Two new concepts: force and mass.Forces
z A force is a push or pull on an object.
z The concept of force gives us a quantitative description of the
interaction between two bodies or between a body and its
environment.
z Some types of forces include:
z Contact forces
z Long-Range Forces (gravitational attraction, electrostatic
force,...)
z Forces are vectors - they have magnitude and direction.
z Forces obey the superposition principle: the effect of any
number of forces applied to a point on an object is the same
as the effect of a single force equal to the vector sum of the
forces applied at that point. Forces
2 1 F F R
r r r
+ =
z Forces are vectors - they have magnitude and direction.
z Forces obey the superposition principle: the effect of any number
of forces applied to a point on an object is the same as the effect of a
single force equal to the vector sum of the forces applied at that
point. Newton’s First Law
z Newton's First Law (Law of Inertia)
z An object at rest will remain at rest unless it is acted upon by
a net external force. An object in motion with constant
velocity will continue to move with constant velocity unless it
is acted upon by a net external force.
z Inertia
z The tendency of a body to keep moving once it is set in
motion.
z Equilibrium
z If the net external force on a body is zero it is said to be in
equilibrium. An object in equilibrium will either be at rest or
in motion in a straight line with constant velocity.
∑ = 0 F
r
∑ ∑ = = 0 0 y x F FNewton’s Second Law
What happens if the net force is not zero?Newton’s Second Law
Experiments show that for any given body the magnitude of the
acceleration is directly proportional to the magnitude of the
net force acting on the body.
m constant= =
∑
a
F
r
r
a m r r
⋅ = ∑F
m is inertial mass of a bodyNewton’s Second Law
2
m  =  k New g  ton N
s
()
Fma =
⎛⎞
⎜⎟
⎝⎠
Can you think of a force that is equal to 1 Newton?
Hint:  Consider the “weight” of a common object
2
1N
1 N 0.1 kg
10 m/s
g Fmg
m
= =
==Newton’s Second Law
If a net external force acts on the body, the body accelerates.
The direction of acceleration is the same as the direction of the
net force. The net force vector is equal to the mass of the body
times the acceleration of the body.
a m r r
⋅ = ∑F
x x F a m⋅ = ∑
y y F a m⋅ = ∑
z z
F a m⋅ = ∑Newton’s Second Law
z The design of high-
performance motorcycles is
based on Newton’s
Second Law
z To maximize the forward
acceleration, motorcycles
are made as light as
possible (minimum mass!)
and have the most powerful
engine possible (maximum
forward force!)
Powerful engine
Large F
Lightweight body
Small m
a m r r
⋅ = ∑F
MAX MAX
MIN MINUnit of Force
z SI unit of the magnitude of force: Newton [N], unit of force that
gives acceleration of 1 m/s2 to a body with a mass of 1 kg.Types of Forces
Contact force: Two objects pushing against each other
Fa,b =  “force acting on a due to b”
Fhead,thumb = “force on head due to thumb”
r
m1 m2
F12 F21
Action at a distance:  Gravitational force, Electromagnetic force
12
12 2
Gravitational Force

mm FG
r
=z Weight of a body: the force of Earth’s gravitational attraction to
a body.
z Weight is a vector!
z Weight acts on bodies all the time, whether they are in free fall
or not.
z Mass characterizes inertial properties of a body.
z Large stone
z Hard to throw because of its large mass
z Hard to lift because of its large weight
Mass and Weight
g w r r
⋅ = mWarm-Up: Newton’s Laws
z N1L: An object at rest will remain at rest unless it is acted upon by a
net external force. An object in motion with constant velocity will
continue to move with constant velocity unless it is acted upon by a
net external force.
z N2L: If a net external force acts on the body, the body accelerates.
The direction of acceleration is the same as the direction of the net
force. The net force vector is equal to the mass of the body times the
acceleration of the body.
z N3L: For every action there is an equal and opposite reaction.
∑ = 0 F
r
∑ = a m F
r r
A on B B on A F F
r r
− =Warm-Up: Newton’s Laws
z N1L and N2L apply to a specific body.
z Decide to which body you are referring! It is not trivial
sometimes.
z Only forces acting on the body matter.
z To analyze person walking, include the force that the ground
exerts on the person as he walks, but NOT the force that the
person exerts on the ground.
z Free-body diagrams are essential to help identify the relevant
forces.
z Action-reaction pair NEVER appear in the same free-body
diagram.
z When a problem involves more than one body: take this problem
apart and draw a separate free-body diagram for each body.Newton’s Third Law
z Newton's Third Law
z Forces always occur in equal and opposite pairs.
z If object A exerts a force on object B, then object B
will exert an equal and opposite force on object A.
z Newton's Third Law is often commonly stated "For every action
there is an equal and opposite reaction."
z The Action and Reaction forces in Newton's Third Law act on
different objects, never on the same object. Force Diagrams
FR on M = FM on RFree Body Diagrams
z Free-Body Force Diagram is a diagram that shows a single object
(as a point) by itself, free of its surroundings, with vectors drawn
to show the magnitudes and directions of all forces exerted ON
the object by other objects.
z Be careful to include only the forces acting ON the object.
z Do not include any forces exerted BY the object on other objects
or on itself.
z Two forces which constitute a Newton's Third Law Action-Reaction
Pair NEVER appear on the same force diagram since these forces
always act on different objects.
z When the problem involves more than one object you should
draw separate force diagrams for each object. Free Body Diagrams
z In some circumstances when objects that are in physical contact are
moving as one unit (both have the same acceleration) it is
acceptable, and in fact useful, to draw a composite force diagram for
the objects. In this case the forces exerted on one object by the other
do not appear on the diagram because the forces would occur in equal
and opposite pairs by Newton's Third Law and would thus cancel each
other.
z You should be able to answer the question "What other body is
applying this force?" for every force on your force diagram. If you
can't answer that question you may be dealing with a non-existent
force.
z Never include non-existent forces such as "the force of acceleration
(the "ma" force)" Newton’s 2nd
Law says that for an object: F = ma . We must isolate the
forces acting only on the object and draw the Free Body Diagram.
THEN, solve for the net force on the object to find its acceleration.
(board not moving!) 0 
0 
xx
yy
Fma
Fma
==
==
∑
∑
B = board
F = floor
W = wall
E  = earth
FW,B  FB,W
FB,F
FB,E
FF,B
FE,B
All Force Pairs
FB,W
+ FB,F
+ FB,E
= 0
FB,W
FB,F  FB,E
Free Body Diagram
Free Body DiagramsApplying Newton’s LawsApplying Newton’s Laws
z We know three Newton’s Laws of motion, the foundation of
classical mechanics.
z We know concepts of forces and masses.
z We know how to draw force diagrams (including free body
diagrams).
z Let’s improve problem-solving skills of applying Newton’s
laws to different real-life situations.Warm-Up: Newton’s Laws
z N1L: An object at rest will remain at rest unless it is acted upon
by a net external force. An object in motion with constant velocity
will continue to move with constant velocity unless it is acted
upon by a net external force.
z N2L: If a net external force acts on the body, the body
accelerates. The direction of acceleration is the same as the
direction of the net force. The net force vector is equal to the
mass of the body times the acceleration of the body.
z N3L: For every action there is an equal and opposite reaction.
∑ = 0 F
r
∑ = a m F
r r
A on B B on A F F
r r
− =Warm-Up: Newton’s Laws
z N1L and N2L apply to a specific body.
z Decide to which body you are referring! It is not trivial
sometimes.
z Only forces acting on the body matter.
z To analyze person walking, include the force that the ground
exerts on the person as he walks, but NOT the force that the
person exerts on the ground.
z Free-body diagrams are essential to help identify the relevant
forces.
z Action-reaction pair NEVER appear in the same free-body
diagram.
z When a problem involves more than one body: take this problem
apart and draw a separate free-body diagram for each body.N1L. Equilibrium
z Body is in equilibrium when it is at rest or moving with constant
velocity in an inertial frame of reference.
z Hanging lamp
z Suspension bridge
z Airplane flying at constant speed
z N1L: When a particle is at rest or is moving with constant velocity
in inertial frame of reference, the vector sum of all forces acting on
it must be zero:
∑ = 0 F
r
∑ ∑ = = 0 0 y x F F
Particle in equilibrium,
vector form
Particle in equilibrium,
component formN1L. Equilibrium of Particle
Problem-Solving Strategy
„ IDENTIFY the relevant concepts: You must use Newton’s first Law
for any problem that involves forces acting on a body in equilibrium.
Remember that “equilibrium” means that the body either remains at
rest or moves with constant velocity. For example, a car is in
equilibrium when it’s parked, but also when it’s driving down a straight
road at a steady speed.
„ If the problem involves more than one body, and the bodies interact
with each other, you’ll also need to use Newton’s third Law. This law
allows you to relate the force that one body exerts on a second body to
the force that the second body exerts on the first one.
„ Be certain that you identify the target variable(s). Common target
variables in equilibrium problems include the magnitude of one of the
forces, the components of a force, or the direction of a force.N1L. Equilibrium of Particle
Problem-Solving Strategy, continued
„ SET UP the problem using the following steps:
1. Draw a very simple sketch of the physical situation, showing dimensions and
angles. You don’t have to be an artist!
2. For each body that is in equilibrium, draw a free-body diagram of this body.
For the present, we consider the body as a particle, so a large dot will do to
represent it. In your free-body diagram, do not include the other bodies that
interact with it, such as a surface it may be resting on, or a rope pulling on it.
3. Now ask yourself what is interacting with the body by touching it or in any
other way. On your free-body diagram, draw a force vector for each interaction.
If you know the angle at which a force is directed, draw the angle accurately and
label it. A surface in contact with the body exerts a normal force perpendicular to
the surface and possibly a friction force parallel to the surface. Remember that a
rope or chain can’t push on a body, but can only pull in a direction along its
length. Be sure to include the body’s weight, except in cases where the body has
negligible mass (and hence negligible weight). If the mass is given, use  w=mg
to find the weight. Label each force with a symbol representing the magnitude of
the force.N1L. Equilibrium of Particle
Problem-Solving Strategy, continued
„ SET UP, continued:
4. Do not show in the free-body diagram any forces exerted by the body on any
other body. The sums in equations include only forces that act on the body.
Make sure you can answer the question “What other body causes that force?” for
each force. If you can’t answer that question, you may be imagining a force that
isn’t there.
5. Choose a set of coordinate axes and include them in your free-body
diagram. (If there is more than one body in the problem, you’ll need to
choose axes for each body separately.) Make sure you label the positive direction
for each axis. This will be crucially important when you take components of the
force vectors as part of your solution. Often you can simplify the problem by your
choice of coordinate axes. For example, when a body rests or slides on a plane
surface, it’s usually simplest to take the axes in the directions parallel and
perpendicular to this surface, even when the plane is tilted. N1L. Equilibrium of Particle
Problem-Solving Strategy, continued
„ EXECUTE the solution as follows:
1. Find the components of each force along each of the body’s coordinate axes.
Draw a wiggly line through each force vector that has been replaced by its
components, so you don’t count it twice. Keep in mind that while the magnitude
of a force is always positive, the component of a force along a particular
direction may be positive or negative.
2. Set the algebraic sum of all x-components of force equal to zero. In a separate
equation, set the algebraic sum of all y-components equal to zero. (Never add x-
and y-components in a single equation.) You can then solve these equations for
up to two unknown quantities, which may be force magnitudes, components, or
angles.
3. If there are two or more bodies, repeat all of the above steps for each body. If
the bodies interact with each other, use Newton’s third law to relate the forces
they exert on each other.
4. Make sure that you have as many independent equations as the number of
unknown quantities. Then solve these equations to obtain the target variables.N1L. Equilibrium of Particle
Problem-Solving Strategy, continued
„ EVALUATE the answer:
o Look at your results and ask whether they make sense.
o When the result is a symbolic expression or formula, try to think of
special cases (particular values or extreme cases for the various
quantities) for which you can guess what the results ought to be.
o Check to see that your formula works in these particular cases.N2L. Dynamics of Particles
z If an object is not in equilibrium then it will experience an
acceleration. The relationship between the net force acting on the
object and its acceleration is given by Newton's Second Law.
x x ma F = ∑ y y ma F = ∑
a m F
r r
= ∑Problem-Solving Strategy
z Make a sketch of the problem showing all objects and
surfaces involved.
z In an area away from your overall sketch, isolate EACH
object in the problem by drawing a free-body force
diagram for the object. This diagram should show all of the
forces acting on the object. Identify each force with an
appropriate UNIQUE symbol.
z Select a convenient coordinate system for each force
diagram and indicate the coordinate axes on your
diagram(s).
z Write each force in component representation (either using
unit vectors or using column vector form). Express
magnitudes as positive quantities and use + or - signs to
indicate directions. Frictional Forces
z Friction force is a contact
force
z Very important force in
everyday life
z Car: tires, brakes, oil in
the engine
z Air drag: parachutes,
resisting motion – bad
fuel economy
z Everything is in place
due to friction: nails,
bulbs, caps…
z Sports: ice hockey,
bicycles,…Frictional Forces
z Static Friction
z Opposes the impending motion.
z Takes on whatever value is necessary to keep the object at rest
(up to a limit), fs < μsN.
z When on the verge of slipping the static frictional force takes on
its maximum possible value of fs, max = μsN.
z μs, the coefficient of static friction, depends on the surfaces in
contact.
z Kinetic Friction
z Opposes the motion of an object that is moving.
z Has a value given by fk = μkN where μk, the coefficient of kinetic
friction, depends on the surfaces in contact.
z It is typically harder to cause an object to move than to keep it
moving, μk < μs. Coefficients of Friction

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