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Chapter 5: The Laws of Motion Chapter 5 Goals: • to introduce the concept of inertia and its relationship to changes in motion • to show how the net force is the physical cause of changes in motion • to present and understand Newton’s First and Second laws of motion • to delve into the weight force • to discuss Newton’s Third law • to build intuition about two more important force examples: the tension force and the normal force • to work with both kinds of the tricky friction forces (kinetic and static friction) Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Newton’s First Law Also called the law of inertia “An object in motion remains in motion, and an object at rest remains at rest, unless a net force acts on the object” Not a quantitatively useful result, but it captures an incredibly insightful observation about motion Until Isaac Newton, most people assumed that motion would cease unless the force continued to act, but this was because ‘friction’ is ubiquitous If an observer sees motion that obeys N1 (again, with no forces present), the observer is said to be in an ‘inertial frame of reference’ Example of non-inertial reference frame: the bed of an accelerating pickup truck Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Some of the Forces in Nature The Spring Force?? The Tension Force!! The Normal Force!! The Weight Force!! The Electric Force!! Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Impulse Force!! The Magnetic Force!! Relationship of Mass to Weight weight force Fg is the gravity force on a body of mass m mass m is the amount of matter in a body [m] = kg, slug W mg and the direction is toward the center of the Earth g = 9.81 m/s2, so a 1 kg mass has a weight of 9.8 N g = 32.2 ft/s2, so a 1 slug mass has a weight of 32.2 lb • if an object ONLY feels the gravity force, then it is in free-fall, which means that it is accelerating downward at g (whether moving up or down!). •If the object is YOU, you would say you were ‘weightLESS’ but in fact what you lack is a force to counteract W. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Weight Force and Tension Force • Lamp is object • There are two forces acting on the object, which are depicted as vectors here: • Tension force (from chain) is UP and therefore POSITIVE • Gravity force (from Earth) is DOWN and therefore NEGATIVE • By N1, if the two forces add to zero, the lamp is either motionless or moving at constant velocity (its Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. acceleration is zero!) Newton’s Second Law Serves to define the mass of the object “When a net force acts on a body, the body accelerates in the direction of the net force. The acceleration is directly proportional to the force, and inversely proportional to the body’s mass” Fnet a or Fnet ma Fnet : F m SI: [a] = m/s2 [m] = kg [F] = kg-m/s2 := N(ewton) A 1 N force causes a 1 kg mass to accelerate at 1 m/s2 USA: [a] = ft/s2 [m] = slug [F] = slug-ft/s2 := lb (‘pound’) A 1 lb force causes a 1 slug mass to accelerate at 1 ft/s2 All of this so far is essentially in one dimension… Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. What is a force? “When a net force acts on a body, the body accelerates in the direction of the net force. The acceleration is directly proportional to the force, and inversely proportional to the body’s mass” Often you literally feel them: when an object presses on you, your biology can sense it Usually: but from the moment you came to life you have felt gravity, and it acts at all points in your body, so you are NOT aware of it until it goes away What is a force? A force is a push or a pull. Enuf said? NO, but that definition has survived and works really well. It is a tautology: anything that is a push or a pull must be a force Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Weight Force and Normal Force • Monitor is object • There are two forces acting on the object: • Normal force (from desk) is UP and therefore POSITIVE • Gravity force (from Earth) is DOWN and therefore NEGATIVE • By N1, if the two forces add to • Book calls the normal force Ftm, zero, the monitor is either for force of table on monitor motionless or moving at •Book calls the weight force FEm, constant velocity (its for force of Earth on monitor acceleration is zero!) Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Third law: involving multiple players • all forces are ultimately caused by bodies: the source of the force is some body • N3: “If the first body exerts a force on a second body, the second body exerts an equal (in magnitude) but opposite (in direction ) force on the first body” • the two forces do not act on the same body!! • Example: a baby of mass m sits in a chair of mass M, which is sitting on the floor • find all N3 pairs and express things precisely as vectors!! Use the unit vector j Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Remarks on T and n • the direction of n is by definition always ‘normal’ to the surface • if a surface exerts a non-normally directed force on a body, it is usually treated as some kind of friction— or GLUE! • often, the normal force magically adjusts itself to suit what the kinematics is doing! • if a weight is supported by a surface, the normal force is equal to the weight in magnitude—but ONLY if the weight is not accelerating! • the contact force between two objects is partly a normal force, and also may include friction Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Remarks on T and n • the direction of T is always ‘along’ the string or rod • T lives without change in all parts of the string or cable that is ‘under’ tension • T can go around corners by the use of pulleys • if a weight is tied to a string and it hangs, the tension is the weight—but ONLY if the weight is not accelerating! • often, the tension magically adjusts itself to suit what the kinematics is doing! Example: a 2 kg hockey skate is accelerating vertically upward at 4 m/s2 because a child is pulling upward on its lace. Find the tension and the weight. Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Some Scenarios Combining T, n and Fg Let the weight Fg of the body be 12 N (so what is mass?) At the beginning, the body is on a horizontal surface Tension T is provided by a hand pulling up on the rope The body may or may not be in equilibrium (that is, a may or may not be zero), and there may or may not be a third normal force n T=10 N T=12 N Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. T=15 N Newton’s Second Law ‘higher dimensions’: Forces are now vectors ma Fnet the net force is a vector Fnet Force every force is a vector • Fnet is also called the resultant force • to grapple with solving N2, we will use the component method of processing vectors: the Free Body Diagram Do we have explicit expressions for a force? • weight force: Fg = mg (g is down and |g|=9.81 m/s2) • spring force: usually it is along one of the coordinate directions, so for example Fspr = –i k x • friction force: two types; both proportional to |n| but direction is perpendicular to n: tangent to the surface Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Elementary example A 4 kg box accelerates at 2 m/s2 to the right on a frictionless floor, because a child pulls on the rope with tension T. The rope makes an angle 60° with the horizontal. 4 kg What do we know about forces on the body, and its acceleration? • we know a, in both magnitude and direction • we know Fg , in both magnitude and direction • we know T, but in direction only • we know n, but in direction only • are there any other forces? NO FRICTION! • the problems are to find magnitudes: |T| = T & |n| = n Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The Free Body Diagram Method 1. draw the body as a simple shape: dot, circle, rectangle 2. draw force arrows, length to scale if you can, either tip or tail on the body, in accurate directions [At this stage in the course all forces act at a single point]. 3. choose a 2d Cartesian coordinate system with origin at that point, oriented with one + coordinate along the acceleration direction if you know that. Draw + and – axes for both coordinates. 4. off to the side, I draw a double arrow for the acceleration if anything is known about it. 5. working component by component, invoke Fnet = ma 6. Now let’s apply the FBD method to that example! Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Another example A box of mass m, on a plane at angle q, is tied by a parallel rope to the wall. a) Find |T| and |n| b) If the rope is cut, find a. q What do we know? • for (a) we know a = 0. • we know Fg , in both magnitude and direction • we know T and n, but in direction only • the problems are to find magnitudes: |T| = T & |n| = n • for (b) we know T = 0 and we know all about Fg • we know the direction of a • the problems are to find magnitudes: |a| = a & |n| = n Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Another example: the Atwood’s machine • two masses m1 and m2 • assume m2 > m1 • massless perfectly limp string • massless pulley, no friction • pulley is supported by force S What do we know? m1 m2 • There are in fact 3 bodies to deal with • we know Fg1 and Fg2, in both magnitude and direction • we know T and S, but in direction only • we know a1 & a2 but in direction only • kinematical constraint: we know a2 = – a1 so let them be a1 = a and a2 = – a, respectively • What do we seek to find, therefore?? Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Three FBDs • Take + = up • Same |T| m1 • We use Newton’s Third Law here: to a1 every action there is an equal and opposite reaction T S T m2 a2 Fg 1 Fg 2 T T m1 T m1 g m1a T m1 a g • now to insert accelerations • a1 is +, a2 is – m2 T m2 g m2 a T m2 a g equate T T a m1 m2 g m1 m2 m2 m1 2m1m2 T g results : a g m2 m1 m2 m1 Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Implications of the Atwood’s machine • a is proportional to g but reduced to nearly zero if masses are close: nice for slowing down accelerated motion!! • T is NOT equal to either of the weights unless the two masses are equal: then it is mg (in which case a = 0 anyway) • what about the third FBD? S T T • it says S – 2T = 0 or S = 2T • interesting consequence of pulleys is how, if stacked, they can have 2 or 3 or 4 or more tensions on them • block and tackle for lifting heavy objects! Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The force(s) of friction: kinetic (sliding) • kinetic friction: when a body is in contact with a surface and the surfaces are in relative motion in their mutual plane, there will be kinetic friction fk on the body due to the surface • the force’s direction is in the plane and opposite to the motion • the force’s magnitude is expressed by The Law of Kinetic Friction: fk = |fk | = mk n • roughly speaking, kinetic friction occurs because of the roughness of the surface • the force depends on both materials via the coefficient of kinetic friction mk (a pure number, typically of the order of .1, .2, up to 1.0 or more Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The force(s) of friction: static (no relative motion)) • static friction: when a body is in contact with a surface and the surfaces are attempting to have relative motion in their mutual plane, but fail to move relatively, there is static friction fs on the body • it is as if the body is stuck to the surface! • the force’s direction is in the plane and opposite to the attempted but failed motion • the force’s magnitude can be understood, depending whether the body accelerates or not… subtle! • this force is somewhat magical, too: it adjusts itself to suit the situation… but only up to a point • its maximum magnitude is ms n so fs = |fs | ≤ ms n • this is The Law of Static Friction Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The force(s) of friction: breakaway (static to the max) • breakaway: when there is no relative motion (due to static friction) but the body is on the verge of ‘breaking away’ into relative motion, the resistive friction has reached the limiting friction fs,max • its direction is in the plane and opposite to the motion • its magnitude is expressed by fs,max = ms n • coefficient of static friction depends on both materials •ms is a pure number, also typically of the order of .1, .2, up to 1.0 or more • Usually, ms > mk ; otherwise things get weird! Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Another example: static friction A box of mass m, on a plane at angle q, is prevented from accelerating by the resistive force. Find fs and n. • we know a = 0; we know Fg • we know fs and n, but in direction only FBD x : mg sin q f s 0 f s mg sin q y : mg cos q n 0 n mg cos q Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. q +y +x fs q F g n Another example: limiting friction A box of mass m, on a plane at angle qB, is on the verge of breakaway. Find ms and n q • we know a = 0; we know Fg • we know fs,max and n, but in direction only +y +x F s,max FBD x : mg sin q B f s,max 0 f s ,max mg sin q B y : mg cos q B n 0 n mg cos q B but Fs ,max m s n mg sin q B m s mg cos q B dividing, we get that m s tan q B nice!! Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. qB F g n Another example • box of mass m, on a plane at angle q, is tied by a horizontal rope to the wall. • find T and n . q The wet diaper problem q q • wet diaper has weight Fg • clothesline droops at angle q • the body is the junction point show that tens ion is T Fg 2 sin q you cannot prevent droop without breaking the clotheslin e Copyright © 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Fg