Mystery of the Physical World
Mystery of the Physical World ISP 209
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This 13 page Class Notes was uploaded by Ms. Mckenzie Labadie on Saturday September 19, 2015. The Class Notes belongs to ISP 209 at Michigan State University taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/207734/isp-209-michigan-state-university in Integrative Studies Physical at Michigan State University.
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Date Created: 09/19/15
To Feel a Force Chapter 2 Spring 2002 Spring forces Chapter 2 Spring forces A The properties of an ideal spring In this chapter a language and a notation to describe all forces is developed from the behavior of elastic forces The relationship governing elastic objects and their forces W s rst codi ed by Robert Hooke and is therefore called Hooke s law An elastic object will distort stretch or compress by an amount that is proportional to applied force Most solid objects behave elastically for small applied forces Elastic objects return quickly to their normal shape when distorting forces are removed Distorting forces that are too large however can cause a permanent change in shape Objects that do not return to their natural shape have been distorted beyond their elastic limit Before the details of how to apply Hooke s law are presented the general features of the forces applied to and generated by a spring will be investigated For now consider only objects that can be represented by an ideal spring a light ie massless but strong coil spring a spring from a retractable ballpoint pen or from a mattress are accessible examples Elastic objects with signi cant mass retain many of these properties they are considered later in this text to simplify the discussion What is not stated in Hooke s law is that the distorting forces must be applied to both ends of the ideal spring Remove a spring from a ballpoint pen and hold it at only one end with the other end free not held or attached to another object See how far you can stretch it No matter how hard you attempt to pull or push on the spring the spring cannot be distorted by any signi cant amount Pushing with an open hand on one end F Entire ideal spring moves rapidly to the right Figure 21 An attempt to apply different forces to the ends of an ideal spring will fail leaving the other end free causes the spring to leave your hand rapidly without compressing as shown in Figure 21 On the other hand if you put two opposing forces of equal strength on opposite ends of the spring you can maintain the spring fixed in front of you with a visible change in length As demonstrated by this experiment and explored further in the next section forces applied to ideal springs must conform to three rules a Forces cannot be applied to only one end of an ideal spring b An ideal spring will distort if and only if forces are applied to both ends and c Forces applied to an ideal spring must be opposing forces of equal strength To Feel a Force Chapter 2 Spring 2002 Spring forces B Action and Reaction Forces In the following discussion consider only springs that are strong light massless and elastic A slinky looks like a spring but it is not elastic After distorting forces are removed it does not return to its original shape From the experimental evidence of the previous chapter it appears that two opposing forces of equal strength one acting at each end are needed to maintain an ideal spring in either a stretched or compressed state as is F F Compressed Stretched F Figure 22 A spring in compressed natural and stretched states and the forces as applied by the hands of a person necessary to maintain those states shown in Figure 22 The compressed state top is obtained by pushing in on both ends of the spring a natural state middle with no forces acting on the spring and a stretched or elongated state bottom obtained by pulling on the ends of the spring The forces generated by a person s hand are represented as an arrow attached to the spring at a point marked by the black dot Allowing some artistic license the hands are removed in Figure 23 to show WW7 Compressed F Figure 23 A spring held by forces vectors with magnitude F in compressed natural F 0 and stretched states only forces acting on the springs Any source applying these forces to the spring will distort it in the same way In contrast the object on which the forces act cannot be removed aforce must always act on an object You must push or pull on something To Feel a Force Chapter 2 Spring 2002 Spring forces The arrows are called force vectors The arrow points in the direction that the force acts with the head or tail touching the object where the force is applied The strength of a force vector is its magnitude speci ed here in two ways a with an algebraic symbol the italic letter F having a precise positive value equal to the strength of the force and b with an arrow drawn with a length proportional to the strength of the force The vector arrows are not drawn with great precision but the relative size eg large medium or small can be useful as a descriptive tool The two force vectors acting on a spring in Figure 23 have the same magnitude F the same length but act in different directions one to le and one to right The two force vectors therefore are not equal to each other the force magnitudes are the same but the force directions are different Two force vectors are equal if and only if they have the same magnitude and direction The forces shown in Figure 23 pull on the spring from outside its ends If the ends are pushed outward ie away from the center of the spring the spring will also stretch Nevertheless a convention will be used where a force acting on a spring has a vector arrow drawn outside the spring while a force generated by a spring has a vector arrow drawn within it and acting on the object at the spring end Later to display large forces generated by small objects this convention may require a magni ed scale In response or reaction to being stretched or compressed by the applied forces the spring generates the internal forces of compression C and tension T shown in Figure 24 A compressed squeezed spring generates compression forces vectors shown A C Compressed Stretched Figure 2 4 A compressed spring generates compression forces C that push the hands outward A stretched spring generates tension forces T that pull the hands inward labeled C that push outward on the objects that maintain the compression spring A stretched spring generates tension forces vectors shown labeled T that pull inward on the objects that maintain the stretch of the spring The italic letters F C or T label the force vector as in the tension T acts on and have values equal to the magnitude of the vector as in the magnitude T is equal to 17 To Feel a Force Chapter 2 Spring 2002 Spring forces The springs can be removed as shown in Figure 25 to exhibit more clearly the reaction forces acting at a point the black dot on the hands Note that each force acts on a different hand C LE Compressed T g gT a Stretched Figure 25 A compressed spring generates compression forces C that push outward A stretched spring generates tension forces T that pull inward An ideal spring shown in Figure 26 reacts by stretching or compressing in response to external forces applied in opposing directions at the ends of the spring At the same time the spring generates internal reaction forces of tension or compression acting on the hands The reaction forces produced by the spring are tension forces T in the stretched spring and compression forces C in the compressed spring Compressed Stretched Figure 2 6 The action and reaction forces of a stretched and compressed spring The statements above apply to ideal springs in the stationary states shown Transitions from the natural length to the states shown in Fig 26 are assumed to occur very slowly The force that the person uses to move the hands has the same magnitude as the force the hands apply to the spring A natural length ideal spring will instantly stretch by a small amount if the hands are moved a small distance away from the center To make that change the person must make both hands apply a small force F to the ends of the spring The change in the spring s length generates small tension forces T with an equal magnitude but opposite direction to the applied force To Feel a Force Chapter 2 Spring 2002 Spring forces For a stretched ideal spring the force magnitudes F and T will be equal T F and for a compressed spring the force magnitudes F and C will be equal C F The stretched spring somehow seems to know just how far to let the hands separate and just how much reactive force is needed to stop the distortion A spring can t know anything and yet it appears as if the spring makes a decision to generate just the right amount of force This illusion is caused by a feedback mechanism involving the motion of the hands Any tendency for the hands to move to a slightly larger separation by slightly increasing the force applied results in a corresponding increase in reaction forces that resist that motion You can try to imagine a very short time when the forces applied by the person to the hands and the reaction forces of the spring do not have the same magnitude At the end of this time you will nd the force magnitudes are again equalized In fact the two forces have the same magnitude at all times If the stretching is rapid the hands with mass attached at the end of the spring cannot not stop instantly and will continue past the point determined by the force the person applies on the hands For now consider massless springs and motion suf ciently slow that the hands move very slowly and can stop rapidly C Newton s 3rd law The mirrorlike behavior of forces acting at the ends of a spring were generalized by Newton in his 3 d law governing the forces at a point of contact between two objects Between two objects 1 and 2 an action force by l on 2 is paired with a reaction force by 2 0n 1 of equal magnitude and opposite direction For our ideal spring it is natural to associate an action force with one of the external forces acting on the spring while the reaction is the internal tension or compression force generated by the spring at the same end The internal and external forces occur simultaneously and either can be considered an action or reaction force At a contact point between objects an action reaction pair of forces always exists In these examples this occurred whenever a hand touched the end of a spring The forces applied by the two hands should not be considered an action reaction pair because they both act on the spring An action reaction pair of forces must act on dijj erent objects The 3rd Law is true in all states of motion of two objects in contact 1 at rest stationary not moving 2 moving with constant speed and direction or 3 accelerating To Feel a Force Chapter 2 Spring 2002 Spring forces speed or direction changing At a point where a person pulls or pushes on an object forces that obey Newton s 3 d law are generated through distortion of the materials at the interface The response of a slinky or chocolate pudding to a distorting force can be very dif cult to analyze however at the contact point of the distorting force a reaction force exists with the magnitude and direction required by Newton s 3 d law To become familiar with elastic forces and Newton s 3rd laws of forces consider an example that brings together the effects of compression and tension to create a stable combination In the model described in Chapter 1 all solid materials act like a spring under tension and compression as long as these forces are not too large A stiff rod shown stationary and being compressed by two hands in Figure 27 is a very strong spring The rod will react to the applied forces magnitude F by generating compression Com ressed F stiff rod F p Figure 27 A stiff rod compressed by two hands is generating compression forces magnitude C that act on the hands forces magnitude C that act on the hands and resist further compression of the rod Compressed by a distance so small that you d need a microscope to see it the rod can generate these large compression forces A strong spring is shown in Figure 28 stretched by outward forces of the same T T Stretched strong spring Figure 28 A strong spring stretched by two hands is generating tension forces magnitude T that act on the hands magnitude F The spring generates tension forces magnitude T that act back on the hands resisting further stretching Distorted by the same magnitude of force F the Figure 2 9 The internal forces acting in a stretched spring and a compressed rod attached at their ends compressed rod of Figure 27 and stretched spring of Figure 28 are now attached with the rod running through the center of the spring as shown in Figure 29 The spring remains stretched and the rod compressed the hands are no longer 20 To Feel a Force Chapter 2 Spring 2002 Spring forces needed The tension forces of the spring pull inward on the rod to keep it compressed while the compression forces of the rod push outward on the spring to keep it stretched An action reaction pair consisting of one tension and one compression force exists at each end of the combination The action and reaction rod compression rod compression forces are indistinguishable and C C the action force on one is the reaction force of the other The T T spring tension spring tension elast1c forces acting on each Figure 210 Rod compression forces keep the spring stretched ObleCt drawn OUtSIde the while the spring tension keeps the rod compressed objects to conform to the previous convention are shown separately for clarity in Figure 210 The construction of a stringed musical instrument shown in Figure 211 is closely related to the rod and spring above All stringed instruments operate on the same principle a stretched string vibrates at a characteristic frequency determined by the length T gt violin string 4 T tuning 639 SCI CW quot bod nedil C y C Figure 21 Schematic drawing of a stringed musical instrument s parts violin and tension of the string The string lies directly on the neck and body though to vibrate freely there must be a small space created by a bridge The instrument must bend slightly to counteract the offset of the string from the centerline of the instrument but that will be ignored in this discussion The string tension adjusted by a screw on which the string has been wound controls the pitch The string behaves much like a spring except its length remains constant as the tension changes The stretched string pulls inward at both ends with a tension force T A small reduction in length of the neck and body generates large compression forces labeled C in Figure 211 that maintain the string tension D Scalars Vectors and Newton s lst Law Vectors quantities having magnitude and direction are unavoidable in a description of the physical world On the other hand many physical quantities are scalars fully speci ed by a single number perhaps with units The magnitude of a vector is a scalar with a positive value Stated emphatically the magnitude of a vector cannot be negative Thus the quantity 710 lb cannot be the magnitude of a force vector 21 To Feel a Force Chapter 2 Spring 2002 Spring forces Some scalars can have negative values Examples are temperature Temp 710 OF units are degrees Fahrenheit sound intensity I 710 dB the units are decibels and an altitude A 745 ft the units are feet Some arbitrary choice for zero eg sea level for altitude determined the sign of these quantities Many physical quantities such as work and potential energy are scalars that can have either negative or positive values others such as length and kinetic energy are positive scalars and cannot take a negative value Along a straight line one dimension there exist two opposing directions Examples are 1 up and down or 2 left and right etc In one dimension the direction of a vector is speci ed by attaching a sign or 7 to the magnitude ln calculations if you see a negative sign attached to a force vector s magnitude eg 7F the direction of the force vector is negative lt s direction points opposite to and 1800 away from a force vector F that points in the positive direction The magnitude of a vector must be a positive scalar changing only its magnitude can never change the direction of a vector Because all vector magnitudes must be positive a force of 710 lb must be the full vector quantity with a 10 lb magnitude and a negative direction A positive quantity like 10 lb 10 1b can be a force vector in the positive direction or just its magnitude A symbol in bold face type such as F is used to represent a vector quantity It contains two separate parts a magnitude and a direction There are no ambiguities in the statement F 10 1b the magnitude F is 10 lb and the direction is positive Specifying the force vector as F 710 lb has the same magnitude 10 lb but a direction that is negative In handwriting script on a lecture transparency or a homework assignment a force magnitude is written with a letter ie F The full vector quantity is written as a letter with an arrow above F bold and italic letters being too dif cult to consistently reproduce by hand In this notation we can describe a situation where Mr A pulls to the right with the force A A and Ms B pulls to the left with a force B B where to the right is assigned the positive direction and to the left the negative 7 direction Forces A and B acting on separate boxes can be shown as vector B vector A A and if the forces act on the same box shown as BEA To Feel a Force Chapter 2 Spring 2002 Spring forces Acting on the same object the two force vectors A and B are said to balance if the magnitudes A and B are equal In the algebra of vectors balance means that the sum of the two vectors is equal to zero A B 0 It is intuitively obvious that for this equation to hold the magnitudes of the two forces must be the same and the force directions must be opposite Also it can be shown to be true algebraically A B 0 Step I replace vector symbols with a magnitude and a direction A B 0 Step 2 add vector 3 to both sides noting that B 3 0 A 3 Therefore the magnitudes must also be equal A B Two force vectors balance sum to zero if 1 they act on the same object 2 point in opposite directions and 3 have equal magnitudes Fach condition must be satis ed for two forces to balance If the motion of mass is unchanging stationary or moving with constant speed and direction then balanced forces act on it Newton s laws I of forces describe the effects of balanced or forces acting on a mass 1 Motion of a mass remains unchanged only by the action of balanced forces 2 Motion speed or direction of a mass changes by the action of unbalanced forces Unbalanced forces and applications of Newton s 2 1 law are addressed in a later chapter E Relationship between balanced forces and action reaction pairs The tension and compression forces acting where the spring touches the rod are members of an action reaction pair each force of the pair acts on a different object one on the rod and one on the spring Balanced forces on the other hand must act on the same object If two forces do not act on the same object they cannot balance To investigate the relationship between balanced forces and actionreaction pairs the rod and spring as shown in Figure 210 are modi ed by attaching massless hooks to the ends of the rod and attaching the stretched spring to the hooks as shown in Figure 212 The tension forces generated by the spring pull the Figure 212 Tension forces of the spring and the compression forces hOOkS Inward Whlle the a rod act on massless hooks at the end of the rod compression forces of the rod Forces are shown displaced to the side for clarity push these same hooks outward In the gure the forces are shown displaced to either side of the combined spring and rod for clarity though in reality the forces of tension and compression act through the center of each object as in Figure 29 On each hook the tension and compression forces have equal magnitudes T C and act on a single object in opposite directions and are therefore balanced Previously 23 To Feel a Force Chapter 2 Spring 2002 Spring forces a tension and compression force comprised an action reaction pair but now they are balanced forces acting on massless hooks By introducing massless hooks or glue between the two objects in contact this relationship between balanced forces and action reaction pairs can always be found The forces acting on a massless object spring hook string or wire if considered massless will always balance Balanced forces acting on an object cause it to compress or stretch The forces acting on a mass will never cancel For the spring and rod as shown in Figure 212 the forces acting clearly do not cancel they have a considerable effect on the components F The balancing of more than two forces The are a limited number of situations where just two forces are in balance More than two forces acting on a single object are in balance if the sum of those forces called the net force is equal to zero FnetF1F2mFn0 130121 The next example a spring is C C stretched and attached on the midline between the sides of a stationary rectangular frame as shown in Figure 213 The pieces of the frame are not connected to each other Only the tension of the Spring keeps the two ends of the frame in Figure 213 Spring stretched in a frame with stiff ends contact with the upper and lower pieces of the frame The short ends of the frame are very strong and will be pulled inward by the spring tension T The long sides react by generating compression forces C that also act on the frame ends Since all parts of the frame are stationary or moving with constant speed and direction the sum of the forces acting on each end of the frame must balance The two tension forces in the spring do not balance The two tension forces are equal in magnitude and opposite in C direction but they do not act on the same piece of the frame Forces must act on the same object in order to balance Focus your attention on only one end of the frame at a time the forces are just reversed on the other end At each end of the stationary frame by Newton s 1st law the two compression C forces acting in one direction and the single tension force acting in the opposite direction as shown in Figure 214 must Figure 214 Forces actng balance The net effect of the two compression forces C acting on the le end Ofthe frame 24 To Feel a Force Chapter 2 Spring 2002 Spring forces on one end of the frame must balance the effect of the spring tension T also acting on that end To accomplish this balancing the magnitudes of the tension and compression forces must have the relationship T 2C This result can be formally obtained using the vector algebra discussed earlier in this chapter Acting on the left side of the frame are two compression force vectors C C one on the top and one on the bottom and one tension vector T T The balance condition Equation 21 is met if the sum of all the force vectors acting on the left side frame is zero Cmp C bum T 0 now replace each vector by its separated form C CT0 now add 2C to both sides T 2C T 2C This relationship can also be obtained using the forces acting on the frame s right side G Pulleys Machines use pulley wheels to manipulate and exploit forces ldeal pulley wheels are massless and free to rotate without friction on an axle that is xed to a bar The rope or belt is also massless and passes over the pulley as shown in Figure 215 Both ends of the rope and bar are held xed somewhere off to the left Figure 215 A pulley wheel and rope The tension in a rope might change when it passes over a pulley wheel as implied by the subscripts on the tension magnitudes It may surprise you to learn that the magnitude of the tension forces in the top and bottom pieces of rope are the same If the tension is higher on the top it will cause the pulley to rotate counterclockwise This brings rope from the bottom increasing tension there to the top lowering the tension there The turning of the pulley wheel tightens the bottom piece of rope and loosens the top piece of rope thus equalizing the tension This is represented by the equation TT1T2 or in words the A value of the two tensions T1 and T2 are equal to each other and can be replaced in all their occurrences by the common T T value T Figure 216 Pulley axles held apart by a compressed bar The forces in Figure 215 could be generated to the left of the pulley by the situation shown in Figure 216 A belt runs over two pulleys held apart by a bar that 25 To Feel a Force Chapter 2 Spring 2002 Spring forces supports the axles The belt has a tension T while the bar is compressed between the axles This compression generates a force with a magnitude C 2T on the axles and they in turn push on each pulley to maintain the belt tension If the top of the belt is grabbed in the middle and pulled to the right and holding the bar in place the tension forces on either side will become unequal brie y the left side tension will increase and the right side tension will decrease The pulleys turn and the belt tension equalizes A stepbystep description of how the tension equalizes is left as an exercise You may have noticed that gravity did not play role in any of the examples shown Gravity is a special case that must be handled carefully to avoid a few very common misconceptions about the balancing of forces A number of future chapters will be devoted to gravity Chapter Summary 0 Force vector arrows are labeled with an italic letter symbolizing the magnitude and represented approximately by the arrow s length To simplify force vector drawings an object generating a force can be removed but a force must always be shown acting on an object 0 Force vectors are equal if and only if the magnitudes and directions are the same 0 External forces acting outward on a spring will stretch it while external forces acting inward will compress it 0 Tension forces generated internal to a stretched spring pull inward on attached objects Compression forces generated internal to a compressed spring push outward on attached objects Elastic objects will distort by an amount that is proportional to the applied force Hooke s Law A strong but lightweight coil spring is a good approximation to a massless ideal spring A slinky toy is NOT an ideal spring 0 Two forces of equal magnitude and opposing directions balanced forces one at each end are needed to maintain an ideal spring in stretched or compressed state An attempt to create an imbalance in forces acting on a massless object instead will cause it to accelerate rapidly away 0 A vector appears in print as a bold letter eg F or in script as a letter with an arrow above eg S has two parts the magnitude F positive scalar and direction a sign or 7 speci es the direction in one dimension Vectors in one dimension obey the standard rules of algebra 0 Force vectors acting on a single object are balanced if the vector sum of the forces is equal to zero Fnet F1 F2 F 0 26 To Feel a Force Spring 2002 Chapter 2 Spring forces Newton s lSt 1aW Balanced forces Will not change the motion of an object With mass This implies that With balanced forces acting on it a stationary mass Will NOT begin to move and a moving mass Will continue to move at the same speed in the same direction Balanced forces acting on an object generate internal forces of compression or tension Newton s 3 d law actionreaction Acting at the boundary between objects are forces With equal magnitude and opposite direction one force acting on each object These forces do not balance because they do not act on the same object 27
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