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FORCES & NEWTON'S LAWS
About this lab: The great aim and accomplishment of Isaac Newton's laws was the prediction of motion, following Galileo's discerning and experimental description of motion. Newton gave primacy to (vector) acceleration of a body as having the simplest prescriptive recipe, (vector force). (But see alternative and equivalent formulation on picture below.) From acceleration, the velocity, position etc. can be obtained by successive integration. A body's acceleration, multiplied by an inherent and persistent property called inertial mass is equal to the (vector) totality of all forces due to other bodies acting on the body. The body must also exert equal but oppositely directed forces on the other bodies. Newton specified the force acting at a distance between heavenly bodies, allowing calculation of their future (and past) motions. In this specification, another inherent and persistent property was introduced: gravitational mass. Experiment shows an amazing proportionality (equality, in our units), finally explained by Albert Einstein's unification of gravity and spacetime.
Studies in the latter half of the twentieth century refined the force concept into the interaction by specific messenger particles among a small set of fundamental (?) particles.
Print out and complete Newton's Laws Prep. Bring it to class.
References: Cutnell & Johnson Physics 6th ed. > Chapter 1 (Trigonometry, vector components, addition of vectors, rectangular (Fx, Fy) and polar (Force magnitude, Force angle) representations of vectors, conversion between the two representations. > Chapter 4: Newton's laws, inertia and inertial mass, the gravitational force, gravitational mass and weight. > Study coordinate system and force components in Example 12, page 102 (replacing an engine weighing 3150 Newtons, with vector input data given in polar form).
Equipment: PC, Vernier software (Logger Pro and Graphical Analysis), force sensor(s) force table, mass hangers, masses, strings attached to ring
5/06 Part A: Equilibrium in Motion in One Dimension
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Newton's Second law: Aerodynamic Drag Forces F = ma ? This is useful, if we can specify F independently; otherwise not. So, think: Recipe = mass x acceleration. If we measure acceleration vs. time experimentally, we can say we know the force. But this is only useful for prediction if we can infer a general recipe from the observed acceleration (or a set of accelerations under varying circumstances). You will follow this procedure to infer some properties of the aerodynamic force acting on a weighted coffee filter in modified fall under the earth's gravitational influence. Shuttle re-entry depends crucially on aerodynamic drag to dissipate excess energy and reduce velocity for landing. This is achieved by pitching the nose up initially about forty degrees and presenting refractory ceramic tiles to the hot plasma sheath which forms.1 It is important to do it right too shallow and the shuttle may skip into space forever, too steep and it may burn up. The driver of an auto traveling at the speed limit adjusts the accelerator pedal for constant equilibrium) speed, at which the force on the tires equals (in magnitude) the drag force. The whole curve of drag force vs. speed is complex, and fit empirically. In higher velocity regimes, such as for the two previous examples, the drag force is proportional to the square of the velocity (turbulent flow). At low speeds, it becomes linear in speed (viscous drag). From the NASA site http://www.grc.nasa.gov/WWW/K-12/airplane/termv.html ,
1 Some say that this method could never work for higher velocity returns, and that return to a non-flying spherical space capsule will be needed, e.g.for Mars flight
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Note: NASA's r above is the air density (Greek rho) below. which says that, at terminal velocity, (no acceleration <-> equilibrium <-> zero net force) the aerodynamic drag force is equal (in magnitude) and opposite (in direction) to the gravitation force W (weight m*g). It also says that the drag force (equal to weight at equilibrium) is proportional to v 2 (true for turbulent flow at sufficiently high velocities), to area A and to air density , with constant of proportionality 2 {C drag/2} . At v < v term, the drag force < W = mg. so the falling object continues to accelerate.
A non-fundamental force recipe: Aerodynamic drag Aerodynamic drag involves the statistical effects of many collisions between the moving object and air molecules. Ionization may be involved (as in shuttle re-entry). The fundamental force involved is electromagnetic (see A Modern View of Force below). But a useful force law ( = recipe) for the turbulent velocity regime can be predicted theoretically and tested empirically.
2 The 2 produces a volume energy density form, if the drag equation is rewritten as v2 = mg/CAfilter .
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Open Logger Pro 3, Experiments, Sample Movies, Air Resistance, Air Resistance 06.cmbl. Select run 1, start; then run 2. Take the mass of a single coffee filter to be 0.001 kg, and it's diameter to be 0.017 meters. Analyze: Select and Analyze (Linear Fit) the later (~ straight) portion of the two y vs. t curves. The individual slopes give the two terminal velocities for the two different masses. Square these terminal velocities and take the difference. Divide this difference by the difference in masses. Call this mass slope. (This saves time; the result should be the same as the slope of a linear fit to the (two point graph) of terminal velocity squared vs mass.) From the coffee filter slope data, predict your own sea level free-fall terminal velocity at 8000 feet. This depends on the air coefficient of drag Cd and on g. Assume these to be the same at sea level and at 8000 feet; they are built in to the coffee filter slope data, so we don't need to include them. The atmospheric density changes between 1.225 (mean sea level) to ~ 0.960 at 8000 feet (kg/m 3); also your mass and projected area during free fall differ from the coffee filter; you must correct for these differences. We know how to correct from the above terminal velocity formula: vyou2 = [coffee filter mass slope] x [your mass] x [(area,cf)/(area,you)] x [(density,sea level)/(density/8000 feet)]. DON'T FORGET TO TAKE THE SQUARE ROOT TO GET v,you. Use consistent mks units. Assume (area,you) is 2 meter 2 . Input your mass M in kg (1 kg = 2.2 pounds). Convert v,you from meters/second to miles/hour. Show your calculations and list your mass in kg and your terminal velocity in mph on your report. (1 m/s = 2.237 mph). Reports on terminal velocities on the hypertextbook and goingtovegas web sites give values from 50 to 100+ mph. http://www.auf.asn.au/groundschool/umodule2.html#density gives air densities to 53000 feet. http://hypertextbook.com/facts/JianHuang.shtml gives a terminal velocity around 55 m/s (with one outlier), but no information about the altitude, e.g about air density. http://www.goingtovegas.com/kpv-dive.htm states that the terminal velocity is around 120 mph (~ 54 meters/second) between 12000 and 4000 feet (novice + jumpmaster, tandem jump).
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Part B: Equilibrium at Rest in Two Dimensions: A Vector Exercise A force table is shown below. Several cords are attached to a small ring at the center. Each cord goes over a pulley to a hanger from which "weights" (masses) can be suspended. The force with which each cord pulls is equal to the weight hanging from it.
The Force Table The weight of each hanging object (in Newton's) is equal to its gravitational mass (in kilograms) times the earth-surface constant g (9.80 m/s2), or to mass in grams times 0.00980. A corresponding string tension force is produced, which acts in the horizontal (two-domensional) plane on the ring. Keep in mind that the hangers, too, have mass (50 g) that must be included in the calculation. (Neglect cord masses).
Analytical and graphical equilibrium condition determination with Graphical Analysis A Graphical Analysis template Newton's Second law: Unknown Equilibrium Force.ga3 has been constructed to do the trigonometry associated with the given forces (calculation of rectangular force components), and to do the component additions. You need to understand the equilibrium conditions (as many as there are spatial dimensions in the problem), know that force is a vector, and be able to transform a vector back and forth between rectangular (x,y) and polar (r,) coordinates.
You must:
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> Enter in the Page 1 table the masses in kg, along with their directions counter clockwise from + x-axis (angles in degrees). You can choose your own x- axis (the theta = 0 direction). Usually you will choose the x-axis along one of the force directions. For force 1, substitute for m1 and theta1 in the table, etc. If you have no given third mass, enter zeros for m3 and theta 3 . Leave the 0's alone; they make the polygon work. > After you enter the given masses and direction angles, the program multiplies the given masses by g = 9.8 to calculate the corresponding force components in Newtons, applies cosine and sine factors to generate x and y components, and sums (separately) these x and y components. > Now your work begins. Look at the x and y sums. They should be non-zero (unless the given forces happen to produce equilibrium, in which case your work is done.) If not, then you must convert this non-equilibrium net force to a magnitude (F 4) and direction (theta 4'), convert the magnitude to the mass m4 which would produce it (divide by g), reverse the direction (to counter balance the other forces), enter your m4 and theta 4, and check that the sums are now zero and that the polygon on Page 2 now closes.
Analytical Procedure: Magnitude of F 4: sqrt(sum of squares of component sums) (Pythagorean theorem); Mass m4: = F 4/g = F 4/9.8 . Enter this. Direction theta 4 of vector sum of given forces: arctan( F y)/ F x). Check which quadrant. Then reverse this (add or subtract 180 degrees) to find Direction theta 4 of unknown force which will produce equilibrium. Enter this. Check your analytical solution: Final force component sums (xz and y separately) should be 0, and force polygon on Page 2 should close. Try your solution on force table. Don't expect perfection there is friction in pulleys.
Study the graphics below.
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Newton's Second Law.Equilibrium Demo.ga3, Page 1. The first two data columns are manual input, the rest are calculated. The alternating rows of zeroes force the GA connect the points graph option to return to zero in generating the force polygon graph. Note that the ultimate sums sumFx and sumFy are now zero and a fourth vector has appeared in each graph. Initial data produces a vector sum graph (sumFy vs. sumFx) which does not close. If there were no additional force, there would be disequilibrium, and the net (penultimate) sumFx would produce x acceleration, the penultimate sumFy would produce y acceleration (accelerations depending on accelerated mass.) But, this is an equilibrium problem. The needed additional mass must produce a string tension force cancel to the penultimate sumFx and sumFy of all the others (opposite component signs, opposite direction.)
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Newton's Second Law.Equilibrium Demo.ga3, Page 2. Equilibrium Force Vector Addition graph, showing (approximately) true angles (vertical and horizontal scales approximately equal). Closure is the graphical condition for equilibrium; this does not depend on whether vertical and horizontal scales are equal.
In each of the following diagrams a set of masses (grams) is shown. The object is to find that force, referred to as the "missing force", to be added to the given set of forces to produce equilibrium (centering of ring). Quantities shown are masses in grams. Enter in kg! Common factors (e.g., g, mass unit) only change the numbers, but not vector proportions; however, the .ga3 programs enter g = 9.8 to calculate forces in Newtons. 1. First Open and study carefully the completed Newton's Second Law.Equilibrium Demo.ga3 for procedure. Re-read the steps listed above. Then, pick a case or two and use the Graphical Analysis template Newton's Second Law.Unknown Equilibrium Force.ga3 to add the given forces by the method of components, and also graphically.
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Use the penultimate sumFx and sumFy values to predict by calculation the component ((x,y) rectangular coordinates) representation and then the magnitude and direction (|F| ,) representation of the missing force needed to give equilibrium. Calculate and enter the corresponding mass with the angle as m4 and th4 in the first two (manual) data columns on page 1. You don't need to adjust the vertical and horizontal scales for every graph to produce approximately true angles just check for polygon closure.
Experimental check of theory. Make a sketch showing the approximate direction you expect for your missing mass, in order to produce equilibrium. Then set up the known forces and your calculated values for missing mass and angle ( m 4 and th 4 ). If the agreement is reasonable, note on report. If not, recheck all your inputs to your calculations (numerical values and units) , since the correctness of the calculations for the numbers you typed in are already checked graphically (polygon closure) and analytically (x and y force sums ~ 0).
Print and submit Page 2 (graphical solution) for one case only (specify the case; print names and note how well experimental check agrees with your predictions).
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5/06 An Experimental Example of Newton's Third Law
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There is probably not time to reproduce the experimental result shown in the graph below, which involves two identical strain gauges (force sensors) hooked together and tugged apart repeatedly.
Pulls the sensors are not well zeroed. Equal (magnitude) and opposite (direction) forces between two interacting bodies Newton's Third Law
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A Modern View of Force
Born: 4 Jan 1643 in Woolsthorpe, Lincolnshire, England Died: 31 March 1727 in London, England
Note the banknote form of Newton's second law, which can be written F = d/dt(mv) = time rate of exchange of linear momentum, m*v. If m is constant, this is equivalent to F = ma. But, in special relativity, the inertial mass m varies in a known way as an object's speed changes relative to the observer. So, although Newton certainly did not anticipate Einstein's special relativity, his formulation survives its advent. But we will study objects moving at small velocities (relative to light speed), so F = ma is an excellent approximation for us. For more commentary, see http://en.wikipedia.org/wiki/Newton's_laws_of_motion .
Isaac Newton was born in the manor house of Woolsthorpe, near Grantham in Lincolnshire. Although by the calendar in use at the time of his birth he was born on Christmas Day 1642, we give the date of 4 January 1643 in this biography which is the "corrected" Gregorian calendar date bringing it into line with our present calendar. (The Gregorian calendar was not adopted in England until 1752.) Isaac Newton came from a family of farmers but never knew his father, also named Isaac Newton, who died in October 1642, three months before his son was born. Although Isaac's father owned property and animals which made him quite a wealthy man, he was completely uneducated and could not sign his own name.
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Isaac's mother Hannah Ayscough remarried Barnabas Smith the minister of the church at North Witham, a nearby village, when Isaac was two years old. The young child was then left in the care of his grandmother Margery Ayscough at Woolsthorpe. Basically treated as an orphan, Isaac did not have a happy childhood. His grandfather James Ayscough was never mentioned by Isaac in later life and the fact that James left nothing to Isaac in his will, made when the boy was ten years old, suggests that there was no love lost between the two. There is no doubt that Isaac felt very bitter towards his mother and his step-father Barnabas Smith. When examining his sins at age nineteen, Isaac listed:Threatening my father and mother Smith to burn them and the house over them. Upon the death of his stepfather in 1653, Newton lived in an extended family consisting of his mother, his grandmother, one half-brother, and two half-sisters. From shortly after this time Isaac began attending the Free Grammar School in Grantham. Although this was only five miles from his home, Isaac lodged with the Clark family at Grantham. However he seems to have shown little promise in academic work. His school reports described him as 'idle' and 'inattentive'. His mother, by now a lady of reasonable wealth and property, thought that her eldest son was the right person to manage her affairs and her estate. Isaac was taken away from school but soon showed that he had no talent, or interest, in managing an estate. An uncle, William Ayscough, decided that Isaac should prepare for entering university and, having persuaded his mother that this was the right thing to do, Isaac was allowed to return to the Free Grammar School in Grantham in 1660 to complete his school education. This time he lodged with Stokes, who was the headmaster of the school, and it would appear that, despite suggestions that he had previously shown no academic promise, Isaac must have convinced some of those around him that he had academic promise. Some evidence points to Stokes also persuading Isaac's mother to let him enter university, so it is likely that Isaac had shown more promise in his first spell at the school than the school reports suggest. Another piece of evidence comes from Isaac's list of sins referred to above. He lists one of his sins as:... setting my heart on money, learning, and pleasure more than Thee ... which tells us that Isaac must have had a passion for learning. Isaac Newton's life can be divided into three quite distinct periods. The first is his boyhood days from 1643 up to his appointment to a chair in 1669. The second period from 1669 to 1687 was the highly productive period in which he was Lucasian professor at Cambridge. The third period (nearly as long as the other two combined) saw Newton as
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a highly paid government official in London with little further interest in mathematical research. Newton's accomplishments were of astonishingly broad scope. For example, as a sidelight to his fundamental contributions in physics and astronomy, he (in parallel with Liebnitz) invented the mathematical discipline of calculus. The poet Alexander Pope was moved to pen the lines Nature and Nature's laws lay hid in night; God said, Let Newton be! and all was light http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html Prior to the 20th century, the nature of classical gravitation and electrostatics (both 1/r2 laws, point to point) were known from the work of Galileo, Kepler, Newton, and Coulomb. The essential character of classical electro-magnetism (they are a unity, along with the less familiar but also fundamental weak interaction) was known from the work of Faraday, Maxwell, Hertz, Lodge and others. Natural radioactivity had just been discovered through the enormous labor of Marie Curie. Einstein discovered theoretically a profound connection between gravitational mass and space-time with space-time being, not passive, but warped and shaped by its gravitating contents. Predictions of this view have been experimentally verified. In the 20th century physicists studied these and found two additional force types of short range, called the weak and strong. Of all these, gravity is by far the weakest. With the discovery and elucidation of the essential quantum mechanical nature of our universe, physicists associated particle quanta (discrete bundles) with these force messenger carriers of the corresponding interactions among matter particles. The best known is the photon, the electromagnetic force carrier of the electromagnetic interaction between charged particles. The graviton, gravity's force carrier has not been observed, due to the extreme weakness of the interaction. The weak force carriers (W+ -, Z0) have been observed, with the help of particle accelerators; the strong force carriers are gluons, believed to be permanently bound, along with quark building blocks, in composites such as neutrons and protons. Discussion of the force messenger particles requires also discussion of the matter particles the messengers serve. A Standard Model of point matter and force messengers (without gravity, which plays little role in atoms, nuclei etc., but which is attractive only and therefore dominant in the
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large (because there is no cancellation, as there is with Coulomb forces) has been painstakingly constructed, with three matter families. Why three families, and the possible relations among them, are unknown, leading to an attempt called string theory (small but extended objects), to unify further the picture below. This is a highly mathematical subject, with the exact theoretical form not yet known. However, it has been shown that quantum gravity and a graviton force carrier of correct properties must be present in these theories, whereas it the graviton presents great difficulty in the pointparticle Standard Model. This has encouraged string theorists. Note that quarks have fractional charge (in electron charge units). The neutrons and protons of our atoms are made up of three quarks, mesons of two. In the standard model quarks are fundamental, along with electrons and neutrinos (leptons, = light elementary particles) and their analogues of the second and third families. The quark concept, supported by experiment, produces an enormous simplification in the description of the very many observed composite particles.
Particle classes (quarks and leptons in three families) and the force messenger particles between them, according to the point particle Standard Model. There is a corresponding chart of anti-particles identical to the particles in mass, but opposite in other properties such as electric
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charge sign. The host of observed particles (protons, neutrons, pi mesons, etc.) which are not shown above are quark composites.