Download PHY1025F-2014-M02-Newtons Laws-Lecture Slides

Physics 1025F
Mechanics
NEWTON’S LAWS
Dr. Steve Peterson
[email protected]
UCT PHY1025F: Mechanics
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Chapter 4: Newton’s Laws of Motion
Dynamics is the description of why objects move
and the connection between forces and motion
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Why do objects move?
What is the “natural state” of
an object [Aristotle, ~350 BC]?
- In Motion or At Rest?
What if we remove friction
(the idealized case [Galileo,
~1600 AD])?
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Newton’s First Law of Motion
Galileo reasoned that the natural state of an object
(if free of external influences) is uniform motion
with a constant velocity.
Consider an object with no force acting on it. If it
is at rest, it will remain at rest; if it is moving, it
will continue to move in a straight line at a
constant speed.
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Fundamental Forces
Four Basic Fundamental Forces
– Strong nuclear force (holds nucleus together)
– Electromagnetic force
– Weak nuclear force (radioactive decay)
– Gravitational force
Characteristics
– All long-range (fields) forces
– Listed in order of decreasing strength
– Only gravity and electromagnetic in mechanics
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Free Body Diagrams
• Identify the object of interest
– The object whose motion you want to study
• Draw a picture of the situation
– Show the object of interest and all directly-interacting objects
– Choose an appropriate coordinate system
• Name and label all the forces acting on the object of
interest
– Contact and long-range forces
• If the free body diagram is incorrect, the solution will
likely be incorrect
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Examples: Free Body Diagrams
1. A block is dragged uphill by a rope. Identify all forces
acting on the block.
2. Block A hangs from the ceiling by a rope. Another block
B hangs from A. Identify the forces acting on A.
3. A ball, hanging from the ceiling by a
string, is pulled back and released.
Identify the forces acting on it just
after its release.
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Newton’s First Law of Motion
A body at rest will remain at rest, and a body
undergoing uniform motion will remain in
uniform motion, unless a net force acts on the
body.
This is sometimes referred to as the Law of Inertia.
Inertia is the tendency of an object to maintain its
state of rest or of uniform motion in a straight
line.
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Newton’s First Law of Motion
If no net force is acting on a body it is said to be in
“equilibrium”.

F0
• e.g. book on table


N
UCT PHY1025F: Mechanics

W
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Static Equilibrium
An object is in equilibrium
when the net force acting on it
is zero. In component form,
this is
F
F
x
0
y
0
- Does acceleration = 0?
- Object at rest or constant v?
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Newton’s Second Law of Motion
What happens if a net force is exerted on an object?
The acceleration of a body is directly proportional
to the net force acting on the body, and is inversely
proportional to the mass. The direction is in the
same direction as the net applied force.
i.e.
UCT PHY1025F: Mechanics

 F
a
m
or


F  ma
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Newton’s Second Law of Motion

 F
a
m
or


F  ma
Note:
• The mass is the property of the body which resists
acceleration, and it is referred to as “inertial
mass” when measured this way.
• If a 1 kg mass is accelerating at 1 m/s2 it must be
subjected to a force of 1 kg m/s2 or 1 N.
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Free Body Diagrams
• Identify the object of interest
– The object whose motion you want to study
• Draw a picture of the situation
– Show the object of interest and all directly-interacting objects
– Choose an appropriate coordinate system
• Name and label all the forces acting on the object of
interest
– Contact and long-range forces
• Draw and label the net force vector Fnet.
• If the free body diagram is incorrect, the solution will
likely be incorrect
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Example: Free Body Diagram
An elevator, lifted by a cable, is going up at a steady speed.
- Identify the forces acting on the elevator.
- Is T greater than, equal to, or less than W? Or is there
not enough information to tell?
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Dynamics
An object is in motion when the net force acting on
it is not zero. In component form, this is
F
F
UCT PHY1025F: Mechanics
x
 ma x
y
 ma y
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Example: Dynamics
A 10 kg block is pulled across a frictionless horizontal floor
by a rope which makes an angle of 30° to the horizontal.
What is the acceleration of the block if the force exerted on
it by the rope is 5 N?
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Newton’s Third Law of Motion
When body A exerts a force on body B, then body B will
exert an equal and opposite force on A.
i.e.


FAB  FBA
Note: Action and
reaction forces ALWAYS
act on different bodies.
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Example: Newton’s Third Law
A book is placed on a table. Are these two forces action
and reaction forces?
A. YES
B. NO

N

W
W = Force of Earth on Book
N = Force of Table on Book
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Weight: The Force of Gravity
A body undergoing free fall accelerated downwards under
gravity with an acceleration g. From Newton’s 2nd law this
means that the body must be experiencing a “force of
gravity”


Fgrav  mg
We refer to the gravitational force as the “weight”,
i.e.
UCT PHY1025F: Mechanics


W  mg
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Mass
Mass is the measure of inertia of an object. In the
SI system, mass is measured in kilograms.
Mass is not weight:
Mass is a property of an object. Weight is the force
exerted on that object by gravity.
If you go to the moon, whose gravitational
acceleration is about 1/6 g, you will weigh much
less. Your mass, however, will be the same.
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Apparent Weight
Gravity is not a force that you can feel or sense directly.
The sensation of weight (how heavy you feel) is due
to the contact forces pressing against you.
When you stand on a scale, the contact force is the
upward spring force acting on your feet. If you and
the scale are in equilibrium, the scale will read your
weight. If not, it will read your “apparent weight.”
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Apparent Weight


 F  ma



N  mg  ma



N  mg  ma
Let’s define the apparent weight (Wapp) as
the magnitude of the contact force (N) that
supports you.

 
Wapp  mg  a 
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Example: Apparent Weight
A 50 kg student gets in a 1000 kg elevator at rest. As the
elevator begins to move, she has an apparent weight of 600
N for the first 3 s. How far has the elevator moved, and in
which direction, at the end of 3 s?
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Normal Force
The key is understanding the normal force is that it adjusts
to the force applied by the object.
The atomic “springs” that make up the surface
produce the normal force.
The harder the objects bears down on the surface,
the more the normal responds, adjusting itself so the
object stays on the surface
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Example: Normal Force
A 10.0 kg box rests on a table. (a) Calculate the normal
force. (b) A rope is now tied around the box and a 40.0 N
force is applied upward, calculate the normal force. (c)
What happens if a 100.0 N force is applied to the rope?
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Example: Normal Force
A 75 kg skier starts down a 50-m-high, 10° slope on
frictionless skis. What is his speed at the bottom?
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Friction
Consider the application of a small force to a stationary
body on a rough horizontal surface.

a

F
At first the application of small force does not induce any motion.
If we apply the force on the other side of the body it still does not
move. (not directional dependant)
We increase the applied force and the body still does not move.
As we increase the applied force the body will eventually begin to
move. (There is an upper limit on the frictional force)
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Friction
Experiments show that the size of the force needed to just
cause movement of the body depends on:
- the nature of the surface (i.e. smooth / rough)
- the normal force exerted by the surface on the body


F fr (max)   s N
where μs is the static coefficient of friction.
Static = two surfaces are not
moving relative to each other
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Friction
In general for the stationary situation


F fr   s N
Once the body begins to move the frictional force usually


reduces so
F fr   k N
where k is the kinetic coefficient of friction and, usually,
k < s.
Kinetic = two surfaces are
moving relative to each other
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Friction
The static frictional force increases as the applied force
increases, until it reaches its maximum. Then the object
starts to move, and the kinetic frictional force takes over.
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Example: Friction
A 2 kg block is pulled across a rough horizontal
surface (k = 0.2) by a rope which makes an angle of
30 to the horizontal. What is the acceleration of
the block if the force exerted by the rope is 5 N?
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Example: Friction
A 2 kg block slides along a smooth horizontal
surface at 2.4 m/s. It then encounters a rough
section of the surface and travels for a further 1.5 m
before it comes to rest. What is the coefficient of
friction between block and table?
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Interacting Objects
Objects in contact will produce Newton’s third law
action/reaction pairs.
Solving two or more objects interacting via
direct contact forces will require applying
Newton’s second and third laws.
Two objects moving together will experience the
same acceleration.
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Example: Interacting Objects
A 2 kg and a 3 kg block are placed in contact with each
other on a smooth frictionless surface. A 10 N force is then
used to push the two blocks across the surface. What is the
acceleration of the blocks? What force does the 2 kg block
exert on the 3 kg? What would change if swapped around?
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Tension
When a cord or rope pulls on an object, it is said to be
under tension, and the force it exerts is called a tension
force.
Assumptions:
• Ignore any frictional effects of the rope
• Ignore the mass of the rope
• The magnitude of the force exerted along the rope is called the
tension and is the same at all points in the rope
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Tension: Connected Objects
Apply Newton’s Laws separately to each object.
The magnitude of the acceleration of both objects will
be the same.
The tension is the same in each diagram.
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Example: Connected Objects
A block with mass m1 = 4.00 kg and a ball with mass m2 =
7.00 kg are connected by a light string that passes over a
frictionless pulley. The coefficient of kinetic friction
between the block and the surface is 0.300. Find the
acceleration of the two objects and the tension in the
string.
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