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1 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.