Transcript Newton’s Laws of Motion - Wayne State University

PHY 5200 Mechanical Phenomena
Newton’s Laws of Motion
Claude A Pruneau
Physics and Astronomy
Wayne State University
Contents
•
•
•
•
Classical Mechanics
Space and Time - Notations
Mass and Force
Newton’s First and Second Laws: Inertial
Frames
• The third law and Conservation of Momentum
• Applications of Newton’s Second Law in
Cartesian Coordinates
• Applications of Newton’s Second Law in Polar
Coordinates
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Classical Mechanics
• Mechanics is the study of how/why things move.
• The Greeks of the antiquity were the first to think
about this - but their notions were seriously flawed.
• Early Modern development of Mechanics are due to
Galileo (1564-1642) and Newton (1642-1727)
• Alternative formulations of mechanics due to
Lagrange (1736-1813) and Hamilton (1805-1865)
– The Newtonian, Lagrangian, and Hamiltonian formulations of
Mechanics are equivalent.
– These three formulations are regarded as “Classical
Mechanics”.
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20th Century Mechanics
• But what about the 20th Century ?
• Realization of inconsistencies between
Maxwell’s Equations and Galilean Invariance
lead to Special Relativity.
– Discussed in PHY 6200
• Discovery of quantized phenomena lead to
Quantum Mechanics.
– Discussed in PHY 6300/PHY 6800.
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Preamble to Newton’s Laws
• Newton’s Laws required four fundamental
concepts/notions
–
–
–
–
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Space
Time
Mass
Force
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Space (According to Newton)
• We live in a 3 dimensional (3D) world.
• Each point, P, of this 3D world can be labeled by a
position vector, r, which specifies the distance,
and the direction, relative to some arbitrary origin,
O. and reference frame S.
• Introduce the notion of coordinates. E.g. (x,y,z).
z-axis
P
r
z
O
x
y-axis
y
x-axis
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Coordinates and Vectors
• Possibly many types of coordinate systems
– Cartesian Coordinates
– Polar, Cylindrical, Spherical Coordinates
– Etc.
• Various vector representations and notations will be
used.
• Cartesian Coordinates, Unit vectors,
Representations
ˆ yˆ, and zˆ
x,
iˆ, ˆj, and kˆ
r  xxˆ  yyˆ  zzˆ
r  xˆi  yˆj  zkˆ
3
eˆ1, eˆ2 and eˆ3
r  x, y, z 
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r  r1eˆ1  r2 eˆ 2  r3eˆ 3   ri eˆ i
r  r1,r2 ,r3 
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i 1
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Vector Properties
•
•
•
•
•
•
Addition, Subtraction
Multiplication by a scalar
Scalar Product (also called dot product)
Magnitude (norm) of a vector
Vector Product
Differentiation of Vectors
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Vector Addition & Subtraction
•
Given two vectors
r  r1,r2 ,r3 
•
s  s1, s2 , s3 
Addition (Sum)
r
r  s  r1  s1,r2  s2 ,r3  s3 
•
r
rs
Subtraction (Difference)
s
r
r  s  r1  s1,r2  s2 ,r3  s3 
r
•
Null Vector
r r
0  r  r  r1  r1,r2  r2 ,r3  r3   0,0,0
r
s
r
rs
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Multiplication by a Scalar
Given a vector
r  r1,r2 ,r3 
r
Multiplication by a scalar, c
r
cr  cr1,cr2 ,cr3 
r
cr
Example:
r
r
F  ma
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c 1
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Scalar Product
Given two vectors
r  r1,r2 ,r3 
s  s1, s2 , s3 
Scalar Product
r
r gs  r1s1  r2 s2  r3 s3
3
  ri si
r

i 1
 rs cos
s
Magnitude of a vector
3
r
r r
r  r  rgr   ri2
i 1
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r2 r r
r  r  r gr
2
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Vector Product (Cross Product)
Given two vectors
r  r1,r2 ,r3 
s  s1, s2 , s3 
r
rs
Vector Product
 xˆ yˆ zˆ 
r r
p  r  s  det  r1 r2 r3 
 s1 s2 s3 
r
p  r2 s3  r3s2 xˆ  r3 s1  r1s3 yˆ  r1s2  r2 s1 zˆ

s
r
r r r r
p  r  s  r s sin
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Important Properties
• Commutability
r r r
r gs  sgr
r r r
r s sr
• Associability
r r r r r
r  s  q  s  r  q 
• But Note
r
r r
r  s  s  r
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Differentiation of Vectors
• Many laws of Physics involve vectors
• Most of these involve derivatives of vectors.
– Different ways to differentiate a vector
– Subject of Vector Calculus
• Here, we start with time derivatives…
• Example
– Position function of time:
– Velocity
– Acceleration
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r(t)
r
dr
v(t) 
dt
r
dv
a(t) 
dt
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Definition of Derivative of a Vector
• Definition closely related to that of scalar functions.
• For a scalar function:
dx(t)
x
 lim
t  0 t
dt
where x  x(t  t)  x(t)
• Similarly, for a vector
r
r
dr
r
 lim
dt t 0 t
where
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r
r
r (t  t)
r r
r
r  r(t  t)  r(t)
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r(t)
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Some useful vector properties
• Derivative of a sum equals the sum of the derivatives
r
r
d r r
dr ds
r  s   
dt
dt dt
• Derivative of a scalar function x vector function is obtained by
the product rule
r
d r
dr df r
 fr   f  r
dt
dt dt
• For a vector represented in Cartesian coordinates, note that
the unit vectors are considered constant (except when
explicitly stated otherwise), one therefore has:
r
dr dx
dy
dz

xˆ  yˆ  zˆ
dt dt
dt
dt
dx
dt
dy
vy 
dt
dz
vz 
dt
vx 
r
dr
v
 vx xˆ  vy yˆ  vz zˆ
dt
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Time
• Classical (Newtonian) view of time is a single universal
parameter, t, on which all observers agree.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
– Provided all are equipped with accurate clocks properly
synchronized.
• This view is actually incorrect and is modified under the theory
of special relativity.
• In the first part of this course, we will neglect effects associated
with special relativity.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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Measurement of Time
Time Measurement Standards and the definition of “the second”
are based on measurements of the natural resonance frequency
of the cesium atom (defined as 9,192,631,770 Hz),
The US - National Institute of Standards and Technology holds
the record in precision accuracy since 1999. The clock NIST-F1
operates with an uncertainty of 1.7 x 10-15, or accuracy to about one
second in 20 million years.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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Reference Frames
• Description of motion in classical mechanics
involves a choice of a reference frame
(either implicitly or explicitly).
• Involves choices of
– Spatial origin,
– Spatial Axes to label positions
– Temporal Origin to measure time.
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About Reference Frames
• Two reference frames may differ in various ways
– Different time origins
– Different spatial origins or axes orientation
– Relative velocity, or acceleration
• Clever choice of reference frame may greatly
facilitate the solution of a specific problem.
• All reference frames are not physically equivalent.
– The laws of mechanics can however be formulated in a
simple form in inertial reference frames.
– The laws of mechanics are independent of the inertial
reference frame being used.
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Mass & Force
• Concepts central to the formulation classical
mechanics
• Subject of many treatises in physics and the
philosophy of sciences…
• We take a pragmatic approach.
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Mass
• The mass of an object is a measure of inertia,
I.e. its resistance to being accelerated.
• Measurement Unit (SI): kilogram.
• Mass measurements
– Inertial Balance
– Weight (Gravitational) Balance
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Mass Units - Kilogram
• The kilogram or kilogramme,
– symbol: kg
– SI base unit of mass.
• Defined as the mass of the
international prototype of the kilogram
– A chunk of platinum-iridium stored at
the International Bureau of Weights
and Measure (Paris, France).
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
• Notes:
– Only SI base unit that employs a prefix,
– Only SI unit that is still defined in
relation to an artifact rather than to a
fundamental physical property.
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Force
• Informally: A measure of push or pull acting
on an object - to move it.
• Formally: A measure of action required to
move a reference mass with a given
acceleration.
• Measurement Unit (SI): Newton (N)
– Newton defined as the force required to accelerate
a standard kilogram mass at 1 m/s2.
• Measurement Technique
– Acceleration of a mass calibrated object: not too
practical.
– Spring balance.
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Note
• In this chapter, we discuss the notions of
velocity, acceleration, force, mass, etc as
applied to point-like objects, or particles.
• We will discuss later how the laws of motions
can be used for extended objects.
• Description of point-like objects is sufficient to
treat very many interesting, quite practical,
physical cases.
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Newton’s First Law
A body remains at rest or in uniform motion unless
acted on by a force.
or
In the absence of forces, a particle moves with
constant velocity.
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Newton’s Second Law
A body acted upon by a force moves in such a manner that
the time rate of change of its momentum equals the force.
r
dp
F
dt
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F
Vector describing the magnitude
and direction of the force acting on
an object
p
Vector describing the magnitude
and direction of the momentum or
“quantity of motion” of an object.
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2nd Law cont’d
Momentum
r
p  mv
m
Scalar quantity describing the amount of
matter (or inertia ) of an object.
v
Vector quantity describing the speed and
direction
an object.
r
dr
v
dt
r
Vector quantity describing the distance
and direction an object realtive to a
chosen origin and axes of reference.
The definition of force is complete and precise only when “mass” is defined.
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Common Formulation of Newton’s 2nd Law
Provided the mass, m, of the object considered is constant,
r
r
one can write
dp d
dv
r
F
dt

dt
mv   m
dt
Usual Formulation of Newtons’ 2nd Law.
r
F  ma
where one defines the acceleration vector, a,
as the rate of change of the velocity.
r
dv
a
dt
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Comments/Notations
We introduce the “dot” and “dot-dot” shorthand notations.
r
dr
r
dt
r
d r
r 2
dt
2
Newton’s 2nd laws and various other quantities may then
be written as F  mvr&
r r
v  r&
r r
a  v&
r r&
a&
r
r
r
F  m&
r&
r r
F  p&
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Newton’s Equation as Differential Equation
Newton’s 2nd law is 2nd order vector differential equation.
r&& r
mr  F
In one dimension, it reduces to
x(t)  F(t) / m
Which is integrable
1
x(t)   F(t)dt
m
&
x(t)   x(t)dt
For constant forces, one gets the familiar results
& 
v(t)  x(t)
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F
t  vo
m
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x(t) 
F 2
t  vot  xo
2m
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Newton’s Third Law
If two bodies exert forces on each other,
these forces are equal and in opposite
directions.
r
F12   F21
1
r
F12   F21
2
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F12
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About the 3rd law
• 3rd law is a law in its own right, an actual statement about the
world.
• 3rd law is NOT a general law.
– Applies to central forces, and contact forces
– Does not apply for some velocity & momentum dependent forces.
• E.g. Lorentz force between moving charges - because of the finite
velocity of light.
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Reformulation of the 3rd Law
• Assume two bodies constitute
an ideal, isolated system.
• Assume constant masses
• Accelerations of two bodies in
opposite directions.
• Ratio of accelerations equals to
the inverse ratio of the masses
of the two bodies.
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r
F1   F2
r
r
dp1
dp2

dt
dt
d
d
r
r
m
v


m
v
 1 1  dt  2 2 
dt
r
r
dv1
dv2
m1
 m2
dt
dt
m2 a1

m1 a2
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Mass Measurement Methods
• Measure relative acceleration of object with unknown mass,
and reference object (with unit mass).
a1
 m2
a2
• Comparison of weights using a balance.
– Make use of F= ma & W=mg
– Where g is independent of a body’s mass.
– Assumes the INERTIAL and GRAVITATIONAL masses are
equal.
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Principle of equivalence
• Inertial Mass:
– Mass determining the acceleration of a body under the action of a
given force.
• Gravitational Mass:
– Mass determining the gravitational force between a body and other
bodies.
• The hypothesis that two masses are equal is called “principle
of equivalence”.
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Experimental Verification of the Principle of equivalence
1.
2.
3.
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First test of the principle of equivalence
performed by Galileo (Pisa).
Newton considered the problem (also) using
pendulums of equal length but made of
different materials.
Recent experiments have tested the
equivalence to a few parts in 1012.
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Isolated system & 3rd Law
• It is impossible to actually have an isolated system.
• Yet…
• The 3rd law implies:
d r r
p1  p2   0

dt
r r
 p1  p2  constant
Implies Conservation of Linear Momentum
Believed to be strictly valid under all circumstances.
Essentially used as Postulate/Foundation of Modern Physics.
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Galilean Invariance
• Laws of motion have a meaning only in inertial reference frame.
• A reference frame can be considered inertial if a body subject to
no external force, moves in a straight line with constant velocity
in that frame.
• If Newton’s laws are valid in a given reference frame, then they
are also valid in any reference in uniform motion relative to that
first frame.
• A change of reference frame involving a constant velocity does
not change the equation.
r
r
r
d(v(t)  vo )
dv(t)
Fm
m
dt
dt
Called Galilean Invariance, or Principle of Newtonian Relativity.
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Implication/Comment on Galilean Invariance
• There is no such a thing as “absolute rest”, or “absolute
inertial reference frame.”
• “Fixed” stars reference frames are a good
approximation to inertial reference but NOT an absolute
frame.
• Both space and time are assumed/required to be
homogeneous.
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Equation of Motion for a single particle
• 2nd law for fixed mass:
r
dp
F
dt
r
r
d mv 
dv
r&
&
F
m
 mr
dt
dt
2nd order differential equation.
If F is known, and initial or boundary
conditions are supplied, it can be
integrated to find the particle position vs
time:
r
r  r(t)
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About Forces…
• The force may in general be a function of any
combination of position, velocity, time.
r r
• Generally denoted:
F r, v,t 
• Seek to know r, v as a function of “t”.
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Conservation Theorems
• Conservation theorems can be “derived” from Newton’s
Eqs.
– If Newton’s laws are correct, then the conservations laws are
also correct; and conversely…
• The fact that conserved quantities are indeed observed in
Nature is a confirmation of Newton’s laws.
• The principles of conservation of momentum, angular
momentum, and energy are sometimes presented as “more
fundamental” - possibly because their formulation enable
extension to other theories - but strictly speaking there are
equivalent to Newton’s laws.
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Conservation of Linear Momentum
For a free particle (no net external force acting on the particle):
F
i,external
r&
 p0
i
p  constant
The total linear momentum p of a system of
particles is conserved (constant) if the net
external force acting on the system is zero.
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Corollary
• The linear momentum conservation theorem can be
decomposed along the 3 coordinates (directions).
• Consider a constant vector s.
s  constant

r
s& 0
Consider scalar product with a force satisfying
r
Fgs  0
independent of time.
r r& r
Fgs  pg
s 0
r
pgs  constant
Component(s) of linear momentum in a direction in
which the (net) force vanishes is constant with time.
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Newton’s Law in Cartesian Coordinates
• Newton’s law is a 2nd order differential
equation.
• It is often possible (and sufficient) to express
the position and force in terms of Cartesian,
constant unit vectors
r  xxˆ  yyˆ  zzˆ
F  Fx xˆ  Fyyˆ  Fz zˆ
r&
&
F  mr
Fx  m&
x&
Fy  m&
y&
Fz  m&
z&
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Useful Problem-Solving Technique
1.
2.
3.
4.
5.
Make a sketch of the problem at hand, indicating forces, velocities, etc
Write down given quantities and information
Write down useful equations
Identify what is to be determined.
Manipulate equations to find quantities you seek.
o
6.
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Algebra, Differentiation, and Integration typically required…
Plug in actual values to determine specific answer.
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Example: Block Sliding on an Incline
Given
Initial position x(0)=0
Initial speed v(0)=0
y
O
f
Fx  m&
x& mgsin  f
Fy  m&
y& N  mg cos  0
N
but
r
w  mg
N  mg cos
f  N
m&
x& mgsin  mg cos

x
x  g sin   cos 
x  g sin   cos t
x
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1
g sin    cos t 2
2
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Newton’s Law in Polar Coordinates
• Usage of polar, cylindrical, or spherical coordinates may enable
enable considerable simplification of solution of specific
problems.
• Definition
By construction
ˆ
rˆ
r
y

O
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x
x  r cos 
y  r sin 
r
x 2  y2
  arctan(y / x)
Definition: Unit radial and polar vectors
r
r
rˆ  r
ˆ  rˆ
r
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Newton’s Law in Polar Coordinates (cont’d)
Force decomposition
F  Fr rˆ  Fˆ
Newton’s Law
r
F&
r&
ˆ
rˆ
r
Derivatives of r in polar coordinates

r  rrˆ
ˆ
r  r&rˆ  rr&
 r&rˆ  r&ˆ
r

O

d
&
r&rˆ  r&ˆ  &
r&
rˆ  2 r&&ˆ  r&&ˆ  r&ˆ
dt


r &
r&
rˆ  2r&&ˆ  r&&ˆ  r&2 rˆ  &
r& r&2 rˆ  2r&& r&&ˆ
Special Case: r=constant, Angular velocity:  Angular acceleration: 
r
a&
r& r 2rˆ  rˆ
Newton’s Law
F
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

 m 2 r&& r&&
Fr  m &
r& r&2
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O
Example: An Oscillating Skateboard
N
A half-pipe for skateboarding
Radius R= 5 m.

Coordinates of the skateboard: (r,) with r=R


 m 2 r&& r&&
Fr  m &
r& r&2
F
Fr  mR&2
F  mR&&
r
w  mg
The forces acting on the skateboard are its weight, and the normal force due to the wall.
r r
F wN
Which can be decomposed as follows:
Fr  mg cos   N
F  mgsin 
mg cos   N  mR&2
mgsin   mR&&
N  mg cos   mR&2
g
R
&&  sin 
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C. Pruneau, W.S.U.
51
Example: An Oscillating Skateboard (cont’d)
O

g
sin 
R

Consider “small” oscillation approximation
N
r
w  mg
sin   
One gets
g
R
 
Which may be written
   2  0
2 
with
g
R
This differential equation has general solutions of the form
(t)  Asin(t)  Bcos(t)
The frequency of the motion is f 
7/18/2015
1
2
g
R
C. Pruneau, W.S.U.
The period is T  2
R
g
52
Example: An Oscillating Skateboard (cont’d)
Determination of the constants A & B.
We have:
O

 (t)  Asin( t)  Bcos( t)
&(t)   Acos( t)   Bsin( t)
N
r
w  mg
Assume, for instance, the skateboard is initially at rest at an angle o.
 (0)  o  B
B  o
A0
&(0)  0   A
So the equations of motion are thus
 (t)  o cos( t)
&(t)  o sin( t)
Note again that the period of this motion is independent of the amplitude o
and has a value
T  2
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R
5m
 2
 4.5s
2
g
9.8m / s
C. Pruneau, W.S.U.
53