Transcript ISP209_Lecture_Sept05

The Laws of Motion
…or, Newtonian mechanics
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The Laws of Motion
…or, Newtonian mechanics
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Isaac Newton
Where is this sculpture from?
Newton believed that mathematics can describe nature, accurately.
He solved the premier scientific problem of his day – to explain the
motion of the planets.
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Newton’s Laws of Motion
1. The law of inertia. An object in motion remains
in motion with constant velocity if the net force
on the object is 0.
2. Force and acceleration. If the net force acting
on an object of mass m is F, then the
acceleration of the object is a = F/m. Or, F =
ma.
3. Action and reaction. For every action there is
an equal but opposite reaction.
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Atwood machine with equal weights
Net force = 0
implies
constant velocity.
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Newton’s second law of motion
F=ma
• F = force acting on the moving object, a vector
• m = mass of the moving object
• a = acceleration of the moving object, a vector
If the force is known, Newton’s
second law states how the object
(on which the force is acting)
accelerates:
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F
a
m
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F=ma
an equation of vectors
Understanding vectors
A vector is a mathematical quantity
that has both a magnitude and a
direction.
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Three kinds of acceleration
m speeds up
m slows down
m changes direction
Example: Circular motion has centripetal
acceleration and centripetal force; F = ma.
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Quantitative applications
of F = ma
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Mass is a property of matter =
quantitative measure of inertia.
The standard unit of mass is the kilogram (kg).
A gram is 0.001 kg.
This picture shows the international
prototype 1 kg mass standard. It is a
platinum-iridium cylinder kept in Sevres,
France.
Any physical object has a mass m, which could
be measured against a standard, e.g., using a
balance.
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Force = a push or a pull exerted on one
object by another object.
The standard unit of force is the newton (N).
The newton is defined in terms of more
fundamental units by 1 N = 1 kg m/s2
(consistent with F = m a)
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Acceleration and Force
Given the mass of the object, and given the force
that is acting on it, the acceleration is
F
a
m
If necessary, identify a and F as vectors and take
into account their direction (both in the same
direction!)
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Example. Playing catch with a softball
The trajectory has 3 parts.
free fall
What is the force acting on the ball, during each
part of the trajectory?
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Example Problem
A dragster accelerates from 0 to 60 mi/hr in
6 seconds. Calculate the force on the car if
the mass is 103 kg.
Answer: 4.47 x 103 N
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Two examples of forces
• weight (the force of gravity on an
object)
• string tension (the force exerted
by a taut string)
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Gravity
What is the force acting on the
mass m due to the Earth’s gravity?
F
Solution
g = 9.81 m/s2
If released, the acceleration
of m would be …
a  g downward
By Newton’s second law
the force on m must be …
F  mg
downward
That is, the magnitude, or strength, is mg
and the direction is downward.
By the way, what is the reaction force?
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Weight = the force of gravity
The weight of an object is, by definition,
the strength of the force of gravity pulling
the object downward.
W=mg
force of gravity
newtons
kg
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Example: What is the weight of a 1 kg mass near
the Earth’s surface?
W = mg = 9.81 N
Or, W = 2.21 pounds
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 different!
Mass and Weight
► Mass is an intrinsic property of an object. It is
completely determined by the number and type of
atoms that make up the object. It does not depend
on the environment in which the object is located.
► But weight is different. Weight depends on both
the object itself, and on some other object that exerts
the gravitational force.
So, for example, the mass of an object
would be the same on the moon or the
Earth; but the weight would be different.
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An astronaut would not need a car jack to
change a flat tire on the Moon Buggy.
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Analyze free fall including air friction
The aerodynamic drag force (C) depends on the
size, shape and surface roughness; it is about the
same for both balls.
• The gravitational acceleration (g) is
independent of mass.
• Effect of air is inversely proportional to mass:
heavy --- small effect of air
light --- large effect of air
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String tension
The force exerted by the string, at either end:
• direction is parallel to the string
• magnitude (same at both ends) is called the tension
Example. Suppose a string can withstand string tension 500
N without breaking. What is the maximum mass M that it
can hold suspended in Earth’s gravity?
Answer: 51.0 kg
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Analyze the motion of a pendulum.
F  Fgravity  T
ma F
Solve by calculus.
L
period  2
g
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More example problems
3. A gymnast weighs 100 pounds.
(a) What is her weight in newtons?
(b) What is her mass?
445 N
45.3 kg
4. A car of 2000 pounds moving 30 mi/hr
crashes into a brick wall. The collision lasts
0.3 seconds. Calculate the force acting on the
car during the collision. Express the answer in
both newtons and pounds.
4.05 x 104 N
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9,100 pounds
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Newton’s second law and calculus
F
F  ma or a 
;
m
instantane ous acceleration is
v
a
as t  0 .
t
differential equation
dv F

dt m
Isaac Newton invented calculus to solve the
equations of motion; i.e., to calculate motion for
the force that is acting. Generally, calculus is the
mathematics that describes continuous change.
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Quiz Question
A car accelerates away from a stop sign with
acceleration 0.1 g ( = 0.981 m/s2). The mass of
the driver is 50 kg. What is the force on the
driver? (Be sure to include the unit of
measurement!)
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