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PHYSICS 1001 REGULAR 2001
Lecture 2
Copyright J.B.T.McCaughan March 2001
DYNAMICS
NEWTON’S LAWS OF MOTION
NI: A body at rest remains at rest. A body in uniform motion in a
straight line remains in that motion (state) unless compelled by
some impressed (external) force to change that state.
TWO BIG IDEAS:
1. Bodies stay PUT in the state in which they were left ( rest
or uniform motion). They do not wander off.
STABILITY OF STATE MAKES SCIENCE POSSIBLE. Without it
there would be no predictibility of behaviour.
2. Because they are stable or have DETERMINATION a
FORCE is required to change that state.
INERTIA: Principle of determination (of state). It can’t be
measured.
'That all bodies are movable, and endowed with certain
powers (which we call the inertia) of persevering in their
motion, or in their rest, we only infer from the like properties
observed in the bodies we have seen.' (From Book III of
Newton’s Principia).
IMPRESSED FORCE:
'An impressed force is an action exerted upon a body, in
order to change its state, either of rest, or of uniform motion
in a right line.'
1Copyright:
J.B.T.McCaughan March 2001
The brief commentary reads:
'This force consists in the action only, and remains no longer
in the body when the action is over. For a body maintains
every new state it acquires, by its inertia only. But impressed
forces are of different origins, as from percussion, from
pressure, from centripetal force.' (From Book I of Newton’s
Principia).
COMMENTS:
(a)Text books to-day divide into two camps, one erroneous the
other defective, on the significance of the first law. The erroneous
one holds that the first law is a special case of the second law
(below) and has no significance of its own. The second holds that
the only significance of the first law is that it is necessary to the
definition of an inertial reference frame (below). While this is true
it is of secondary importance. The significance of the first law is
that it defines how the basic condition of nature is to be
understood: two states, straight line motion and stability requiring
an external agent to change. It is not immediately deducible from
nature that this is the only way to start, particularly that there are
two equal basic states of nature. Linked to the law, but not part of
it, is that the stability is to be understood through inertia and not,
for example, through impetus. The first law is often referred to as
the ‘law of inertia’, but inertia is not necessarily deduced from it.
INERTIAL REFERENCE FRAME: One where a body is in a state
of rest or uniform motion in a straight line and is not subject to an
impressed force.
(b) There is now a widespread belief that Newton’s laws are only
valid in an inertial reference frame (see HRW 6th ed p74). THIS IS
FALSE. One is stuck in a circular argument if this is the case: One
needs an inertial reference frame to validate the laws, but one
needs the first law to identify the inertial reference frame. Worse
still, Newton’s laws would not then be laws of nature. They have
restricted validity; what law of nature could there be whose
existence depends on how one observes nature? It then becomes a
function of the observer and not of nature. Further, it would not
only require the observer for its validity (as if nature could only
function if we observed it) it would be founded on the mind of the
observer not on nature since the first step is a choice of the mind.
Besides all that one needs the first law to know that it is
(apparently) violated: an object's motion appears to be accelerated
without the application of an external (impressed) force. Because
Newton's laws are always valid one immediately expects the
impressed force to be on the observer and not on the object (see
the appendix below).
FORCE OF INACTIVITY (vis inertiae): Force elicited from the
body, as a consequence of its inertia, in response to an impressed
force. It has two components: 1. An intrinsic component that limits
the acceleration of the body resulting from the impressed force. 2.
An extrinsic component, called INERTIAL FORCE, directed at the
body of the agent of the impressed force, if the agent has a body.
Part of Newton’s commentary on force of inactivity reads as
follows:
‘But a body only exerts this force when another force,
impressed upon it, endeavours to change its condition; and
the exercise of this force may be considered as both
resistance and impulse; it is resistance so far as the body, for
maintaining its present state, opposes the force impressed; it
is impulse so far as the body, by not easily giving way to the
impressed force of another, endeavours to change the state
of that other’.
The resistance limits the acceleration. Impulse (not to be confused
with the modern term) has been rendered as inertial force (the
anglicised form of vis inertiae) as this term is the one in common
usage.
NII: Mass times acceleration is proportional to the impressed force
and is made in the direction of the straight line in which that force
is impressed.
 F = ma
COMMENTS:
(a) The second law quantifies the second idea in the first law. The
second law’s reason of being lies in the first law.
(b) Newton never wrote the second law that way, nor did he write
down the equation ( which is due to Euler). What he wrote was:
'The change of motion is proportional to the motive force
impressed; and is made in the direction of the right line in
which that force is impressed.'
By ‘motion’ Newton meant what we today call momentum.
Clearly the law is not right as it stands. The majority opinion
among Newton scholars is that by force Newton meant what
today we call impulse (force by time interval). But from later
comments he made in the Principia it can be seen that Newton
clearly had the correct idea as given in our version of the law
above.
STANDARD of Mass (and Force).
In setting up Dynamics an additional standard to length and time
is required. THE law of dynamics is F = ma. Acceleration is
standardised so only one of m or F needs standardisation as the
other is dependent through the formula. The curious thing is that
F and m are defined together but only mass is preserved as the
standard as it is easier to conserve and copy.
The first step is draw the graph of the answer we want and then
do the measurements!
Force/N
m = 1 kg
4
3
2
1
1
2
3
4
Acceleration ms -2
The abscissa is labeled as acceleration for which standards already
exist. Intervals are marked on the axis in the appropriate units
(ms-2). Next a straight line is drawn from the origin at (say) 45
degrees to the axis. This is labeled slope m = 1kg. Then the points
on the ordinate, labeled for force, are marked off corresponding to
the intervals on the abscissa. I.e. use is made of the formula F = ma
for which m =1 kg. The units are Newtons. So 1N = 1kg.1ms-2, 2N
= 1kg.2ms-2 etc.
So far no measurements have been taken.
DEMONSTRATION: Trolley, brass weights and dynamometer.
The trolley for the purpose of demonstration is declared to be the
standard 1kg. The dynamometer is a spring that exerts a force that
varies with the extension of the spring. It is NOT necessary that
the extension be proportional to the strength of the force. The
dynamometer is attached to the trolley and the extension x1 that
results in an acceleration of 1ms-2 is recorded. Similarly x2, x3, etc
that corresponds to 2, 3ms-2 etc. The dynamometer is now
calibrated with 1N = x1, 2N = x2, 3N = x3 etc. Now standards of
force and acceleration exist which may be used to calibrate other
masses.
For example brass weights are added to the trolley increasing its
mass. The standard dynamometer ( literally force meter) is now
used to accelerate it and the coordinate pairs of N and ms-2 are
plotted on the graph. A new straight line passing through the
origin connecting the points is drawn. The (steeper slope) of this
line is now the new mass. This may be repeated for other masses.
The standardisation procedure is complete.
MASS: It has two meanings: 1. Quantity of matter: It is tied to the
number of atoms in a body and is useful in chemistry. It is linked
in an unclear way to its dynamic meaning: 2. Coefficient of inertia:
The ratio between impressed force and acceleration measured in a
reference frame which partakes of the original motion of the body.
The reference frame need not be inertial.
INERTIAL REFERENCE FRAME: (second definition) One where
no inertial forces are generated in the observer or any system to
which the frame is attached.
COMMENTS:
(a) The second definition of an inertial reference frame is useful if
one cannot tell if a body is subject to an impressed force or not. It
is not possible to accelerate a body or an observer composed of
parts without internal stresses being produced by the inertial
force. (The situation of gravitational acceleration is exceptional,
see the appendix below).
(b) Determining the correct extension of the dynamometer and
hence the correct mass in a non-inertial (accelerated) reference
frame is simply a matter of determining the direction of the
inertial force being generated in the accelerated frame and then
choosing to perform the acceleration of the unknown mass by the
dynamometer in the same direction and in the opposite direction,
then averaging the result. The extension of the dynamometer due
to the acceleration of the observer, produced by the inertial force,
drops out leaving the extension it would have had in an inertial
reference frame. This strategy of course relies on the principle of
superposition (each force produces its own acceleration
independently of other forces present) and the linearity of the
spring. This is also the strategy to use when calibrating the
dynamometer from the standard mass at the beginning. If one has
to use a non-linear spring, then the procedure is more complicated
(see the appendix below).
(c) A way of visualising the difference between the two concepts
of mass is to consider the increase in mass in the relativistic
domain (at speeds approaching the speed of light). It is not
necessary to think that the body increases mass by adding matter
and growing bigger in size. Only in its dynamic meaning does it
grow in value; it is harder to push to a higher speed at these
speeds. (Of course the point is moot as to whether the increase
should be assigned to the mass or the inertial force).
(d) Mass is NOT the measure of a body’s inertia, as sometimes
stated in text books. Inertia can’t be measured. What one is
conscious of when accelerating a body is its inertial force acting
back on you or the agent doing the acceleration. Inertial force is
given by:
F ma
Force is not mass. Just as mass is not weight in the gravitational
situation. Force varies with acceleration, mass does not (in the
non-relativistic domain we are restricted to).
(e) Only with inertial force is F ma. In all other cases F = ma.
Force has its own source, its own existence, such as gravity,
electricity etc and its own law such as Newton’s law of gravitation
or Coulomb’s law. Its measure is equal to the measure of mass times
acceleration. The existence of force and the existence of an
accelerated body are separate in nature. Force can exist, say a
spring stretched between your hands, without any acceleration
being present. The measures are equal the existences are not. Even
when acceleration is present, the agent doing the accelerating is
not the object undergoing the acceleration. Rejected is the opinion
of nearly 50% of text books that force is defined as mass times
acceleration.
(f) Acceleration is ABSOLUTE. When determined from a known
force and a known mass it obviously follows from the formula. If
acceleration is measured from a reference frame, the object should
be at rest (or in uniform straight-line motion) in the frame before
the force is applied. If the resulting accelerated motion is straight,
then the absolute value for the acceleration is obtained directly,
corresponding to the applied force. If a sideways component of
motion appears as well, only the component of motion in the
direction of the force is accepted and this too will be absolute for
that applied force.
The above strategies use the vector nature of force. The principle
of superposition underlying vectors ensures that each force
produces its own acceleration; it does not matter if the reference
frame is being accelerated by other forces. The effect of a force is
only in the direction of the force as NII states. Sideways
acceleration, if present, is an effect of the acceleration of the frame.
No inquiry as to whether the frame is inertial or non-inertial need
be made. It should becoming plain that there is no need to identify
the type of reference frame. Newton's laws are force based NOT
frame based.
Only velocity needs a reference frame for its determination.
Velocity is usually RELATIVE. There is no dynamics to break the
symmetry of the kinematical setting since dynamics only involve
accelerations and their associated forces: Either object or observer
is moving with the velocity. In the one case of uniform circular
motion, speed relative to the centre of the circle can be determined
absolutely, because it is obtained from the centripetal acceleration.
(C.f. Book I of Newton’s Principia). There is an ultimate reference
frame in the cosmic microwave background (CMB). Velocity is
determined from this by Doppler shift and is ABSOLUTE. In the
interactions between bodies or systems, only the relative value is
needed.
APPENDIX
(a) Absolute or true (accelerated) motion is associated with force:
'The causes by which true and relative motions are
distinguished, one from the other, are the forces impressed
upon bodies to generate motion. True motion is neither
generated nor altered, but by some force impressed upon the
body moved; but relative motion may be generated or
altered without any force impressed upon the body.'(From
Book I of Newton’s Principia).
Thus, if an observer in a rocket ship sees a similar rocket ship
accelerate away from him/her, then according to kinematics the
acceleration is relative in that it is mutual. Seen from either ship
the other is accelerating.
According to dynamics only the ship subject to external force is
the one that is actually accelerated. Thus it is the other ship that is
accelerated if the observer sees its rocket engines firing or notes
that his/her own ship is not subject to inertial force. The reverse is
true if the observer's engines are fired and/or the inertial force is
felt.
When the external force is gravity, a peculiar situation arises.
Suppose the observer is in a space station orbiting the Earth. Since
everything in this orbiting laboratory is in free fall, to a good
approximation there is no inertial force in the lab. (There is a small
inertial force arising from the gradient of the Earth's gravitational
field across the lab. There is no such thing as a uniform
gravitational field since it sources effectively from a point). To be
in free fall is effectively to be in a field free region of space. This is
the peculiarity. Measured accelerations will still be absolute. They
are the change in state from the state of free fall. Another way of
stating the peculiarity of this situation is that the laboratory is an
accelerated reference frame that passes the test of an inertial
reference frame.
(b) Calibrating the dynamometer that uses a non-linear spring in a
non-inertial reference frame: In the direction of the inertial force
determine the spring extension for the standard mass (held at rest
in the frame). Produce a number of standard masses (checked by
the same extension on the spring). Calibrate extension verses one,
two, three, four, etc, standard masses. Label these as x1, x2, x3, x4,
etc. Suppose the unknown acceleration of the frame is given by g.
Then the extensions correspond to forces 1g, 2g, 3g, 4g, Newton
etc. Now apply the dynamometer to one standard mass to
produce absolute accelerations in the same direction of the inertial
force of a = 1ms-2, 2ms-2, 3ms-2, 4ms-2, etc which must
correspond to forces of 1N. 2N, 3N, 4N, etc and for which the
extensions are y1,y2, y3, y4, etc. This would calibrate the
dynamometer for that particular acceleration of the reference
frame. What we want to do is to calibrate it for an inertial frame.
Now suppose y2 = x4. (The y extensions are greater than x as they
start with x1 and not from zero). Then 4g N = 2N and g is solved.
Then the absolute acceleration in the inertial frame is a+ g. The
dynamometer is recalibrated with y1 now = (1+g) N; y2 = (2+g) N;
y3 = (3+g) N etc.