The Dot Product (Scalar Product) Chapter 9 Statcs and Torque The dot product of two vectors can be constructed by takng the component of one vector n the drecton of the other and multplyng t tmes the magntude of the other vector. Ths can be epressed n the form: The Dot Product (Scalar Product) The Dot Product use (Scalar Product) Geometrcally, the scalar product s useful for fndng the drecton between arbtrary vectors n space. Snce the two epressons for the product: The Cross Product The magntude of the vector product of two vectors can be constructed by takng the product of the magntudes of the vectors tmes the sne of the angle (<180 degrees) between them. The magntude of the vector product can be epressed n the form: The Cross Product the drecton s gven by the rght-hand rule. If the vectors are epressed n terms of unt vectors, j, and k n the, y, and z drectons, then the vector product can be epressed n the rather cumbersome form: 1
The Cross Product (The Matr) Force vs. Torque A B A B A B ˆ a b ˆj a b y y kˆ a b z z a b y z z a b a b y a b z y y a b z a b Forces cause acceleratons Torques cause angular acceleratons Force and torque are related Torque The door s free to rotate about an as through O There are three factors that determne the effectveness of the force n openng the door: The magntude of the force The poston of the applcaton of the force The angle at whch the force s appled Torque Torque,, s the tendency of a force to rotate an object about some as F d s the torque symbol s the Greek letter tau F s the force d s the lever arm (or moment arm) Agan: Torque causes objects to rotate lke force causes object to move. SI Unts Newton meter = Nm Drecton of Torque Torque s a vector quantty The drecton s perpendcular to the plane determned by the lever arm and the force For two dmensonal problems, nto or out of the plane of the paper wll be suffcent If the turnng tendency of the force s counterclockwse, the torque wll be postve If the turnng tendency s clockwse, the torque wll be negatve Rght Hand Rule Pont the fngers n the drecton of the poston vector Curl the fngers toward the force vector The thumb ponts n the drecton of the torque 2
Lever Arm An Alternatve Look at Torque The lever arm, d, s the perpendcular dstance from the as of rotaton to a lne drawn from the as of rotaton to a lne drawn along the the drecton of the force d = L sn Φ What we are dong here s fnd the component of F that s perpendcular to L The force could also be resolved nto ts - and y- components The -component, F cos Φ, produces 0 torque The y-component, F sn Φ, produces a non-zero torque Torque, fnal equaton From the components of the force or from the lever arm, FLsn F s the force L s the dstance along the object Φ s the angle between the force and the object Net Torque The net torque s the sum of all the torques produced by all the forces Remember to account for the drecton of the tendency for rotaton Counterclockwse torques are postve Clockwse torques are negatve Torque and Equlbrum Frst Condton of Equlbrum The net eternal force must be zero F F 0 0 and F y Ths s a necessary, but not suffcent, condton to ensure that an object s n complete mechancal equlbrum Ths s a statement of translatonal equlbrum 0 Torque and Equlbrum, cont To ensure mechancal equlbrum, you need to ensure rotatonal equlbrum as well as translatonal The Second Condton of Equlbrum states The net eternal torque must be zero 0 3
Equlbrum Eample Mechancal Equlbrum The woman, mass m, sts on the left end of the see-saw The man, mass M, sts where the see-saw wll be balanced Apply the Second Condton of Equlbrum and solve for the unknown dstance, In ths case, the Frst Condton of Equlbrum s satsfed F 0 5 0 0 N 5 0 0 N The Second Condton s not satsfed Both forces would produce clockwse rotatons 500Nm 0 As of Rotaton If the object s n equlbrum, t does not matter where you put the as of rotaton (Pvot Pont) for calculatng the net torque The locaton of the as of rotaton s completely arbtrary Often the nature of the problem wll suggest a convenent locaton for the as When solvng a problem, you must specfy an as of rotaton Once you have chosen an as, you must mantan that choce consstently throughout the problem Center of Gravty (Center of Mass) The force of gravty actng on an object must be consdered In fndng the torque produced by the force of gravty, all of the weght of the object can be consdered to be concentrated at one pont In an object where all the mass s unformly dstrbuted, the center of mass s the geometrc centrod. Center of Gravty When suspended from a strng, an object wll always rotate so that ts center of gravty s drectly below the strng. Fg. 5.27, p. 141 Slde 30 4
Center of Mass: An object wll balance when ts center of mass s drectly above ts base of support. Calculatng the Center of Gravty The object s dvded up nto a large number of very small partcles of weght (mg) Each partcle wll have a set of coordnates ndcatng ts locaton (,y) Calculatng the Center of Gravty, cont. The torque produced by each partcle about the as of rotaton s equal to ts weght tmes ts lever arm Calculatng the Center of Gravty, cont. We wsh to locate the pont of applcaton of the sngle force, whose magntude s equal to the weght of the object, and whose effect on the rotaton s the same as all the ndvdual partcles. Ths pont s called the center of gravty of the object Coordnates of the Center of Gravty The coordnates of the center of gravty can be found from the sum of the torques actng on the ndvdual partcles beng set equal to the torque produced by the weght of the object cg m m and y cg m y m 5
Eample: An 8.0-kg mass s poston at = 2.0-m, a 5.0-kg mass s postoned at = 4.0-m, and a 2.0-kg mass s placed at = 12 m. Where s the center of mass of ths system? Soluton: An 8.0-kg mass s poston at = 2.0-m, a 5.0-kg mass s postoned at = 4.0-m, and a 2.0-kg mass s placed at = 12 m. Where s the center of mass of ths system? Center of Gravty of a Unform Object The center of gravty of a homogenous, symmetrc body must le on the as of symmetry. Often, the center of gravty of such an object s the geometrc center of the object. Epermentally Determnng the Center of Gravty The wrench s hung freely from two dfferent pvots The ntersecton of the lnes ndcates the center of gravty A rgd object can be balanced by a sngle force equal n magntude to ts weght as long as the force s actng upward through the object s center of gravty Whch sees more weght? The person has a mass of 80Kg and the board has a mass of 5kg. The persons center of mass s 8/10 of L/2. The person s 1.7m, the board s 2m. Both the board and the person are centered on the scale and the block. Notes About Equlbrum A zero net torque does not mean the absence of rotatonal moton An object that rotates at unform angular velocty can be under the nfluence of a zero net torque Ths s analogous to the translatonal stuaton where a zero net force does not mean the object s not n moton 6
Fg 8.12, p.228 Slde 17 Eample of a Free Body Dagram Isolate the object to be analyzed Draw the free body dagram for that object Include all the eternal forces actng on the object Eample of a Free Body Dagram The free body dagram ncludes the drectons of the forces The weghts act through the centers of gravty of ther objects Notce the components are calculated Try ths Torque Problem (Case I) m1= 5 kg @ 2.0m m2= 10kg @ 0.5m m3=??? @ 2.0m Try ths Torque Problem (Case II) The chld has a mass of 60kg and the unform board has length of 3m and s 5kg. The fulcrum s located 0.5m from the end of the board. How far away from the fulcrum must the chld st? How s ths torque changng wth X? 7
Why does ths bolder stay standng? Eample of Torque n Health Felds Eample of Torque n Health Felds Eample of Torque n Health Felds Torque n constructon Recreatonal Torque 8
Another Recreatonal Torque Try ths Problem (Case III) The guy on the left s 70kg and the guy on the rght s 78kg. The bucket s 5kg and the scaffoldng s 65kg. What s the force n each of the strngs? Torque and Angular Acceleraton When a rgd object s subject to a net torque ( 0), t undergoes an angular acceleraton The angular acceleraton s drectly proportonal to the net torque The relatonshp s analogous to F = ma Newton s Second Law Moment of Inerta The angular acceleraton s nversely proportonal to the analogy of the mass n a rotatng system Ths mass analog s called the moment of nerta, I, of the object I mr SI unts are kg m 2 2 Newton s Second Law for a Rotatng Object The angular acceleraton s drectly proportonal to the net torque The angular acceleraton s nversely proportonal to the moment of nerta of the object I More About Moment of Inerta There s a major dfference between moment of nerta and mass: the moment of nerta depends on the quantty of matter and ts dstrbuton n the rgd object. The moment of nerta also depends upon the locaton of the as of rotaton 9
Moment of Inerta of a Unform Rng Other Moments of Inerta Image the hoop s dvded nto a number of small segments, m 1 These segments are equdstant from the as I m r 2 MR 2 Eample, Newton s Second Law for Rotaton Draw free body dagrams of each object Only the cylnder s rotatng, so apply = I The bucket s fallng, but not rotatng, so apply F = m a Remember that a = r and solve the resultng equatons Rotatonal Knetc Energy An object rotatng about some as wth an angular speed, ω, has rotatonal knetc energy ½Iω 2 Energy concepts can be useful for smplfyng the analyss of rotatonal moton Total Energy of a System Conservaton of Mechancal Energy ( KE KE PE PE ) ( KE KE PE PE ) t r g s Remember, ths s for conservatve forces, no dsspatve forces such as frcton can be present t r g s f Work-Energy n a Rotatng System In the case where there are dsspatve forces such as frcton, use the generalzed Work- Energy Theorem nstead of Conservaton of Energy W nc = KE t + KE R + PE 10
Angular Momentum Smlarly to the relatonshp between force and momentum n a lnear system, we can show the relatonshp between torque and angular momentum Angular momentum s defned as L = I ω L and t Angular Momentum, cont If the net torque s zero, the angular momentum remans constant Conservaton of Lnear Momentum states: The angular momentum of a system s conserved when the net eternal torque actng on the systems s zero. That s, when 0, L L f or I I f f Vector Nature of Angular Quanttes Rght Hand Rule / Screw Rule Assgn a postve or negatve drecton n the problem A more complete way s by usng the rght hand rule Grasp the as of rotaton wth your rght hand Wrap your fngers n the drecton of rotaton Your thumb ponts n the drecton of ω How all rotatonal quanttes are related Precesson Start by thnkng about the object not spnnng. Where would t fall. A wheel held up by the strng on the rght would fall down. Produced by a torque nto the screen (paper) From: http://hyperphyscs.phy-astr.gsu.edu/hbase/rotv.html#rvec2 From: http://hyperphyscs.phy-astr.gsu.edu/hbase/rotv2.html#rvec4 http://hyperphyscs.phy-astr.gsu.edu/hbase/top.html#top 11
Precesson Net thnk about the object rotatng There would be a torque ponted to the rght, out of the wheel Precesson The dfference n these two torques (one down and one to the rght) s lke two forces one at 90 and another at 0. The resultant would be somethng n-between. The sum of these torques s somethng n-between. Snce there s a net torque, the object has an angular acceleraton n the drecton of the torque. From: http://hyperphyscs.phy-astr.gsu.edu/hbase/rotv2.html#rvec4 http://hyperphyscs.phy-astr.gsu.edu/hbase/top.html#top From: http://hyperphyscs.phy-astr.gsu.edu/hbase/rotv2.html#rvec4 http://hyperphyscs.phy-astr.gsu.edu/hbase/top.html#top Conservaton Rules, Summary In an solated system, the followng quanttes are conserved: Mechancal energy Lnear momentum Angular momentum Conservaton of Angular Momentum, Eample Wth hands and feet drawn closer to the body, the skater s angular speed ncreases L s conserved, I decreases, ncreases Machnes A machne s a devce that helps make work easer to perform by accomplshng one or more of the followng functons: transferrng a force from one place to another, changng the drecton of a force, ncreasng the magntude of a force, or ncreasng the dstance or speed of a force. Machnes Machnes do not reduce the amount of work for us, but they can make t easer. There are s types of smple machnes whch form the bass for all mechancal machnes today. 12
(Smple) Machnes The 6 smple machnes are: Lever Inclned plane Wheel and Ale Pulley Wedge Screw Quck Overvew of Each SIMPLE MACHINES LEVER INCLINED PLANE WHEEL AND AXLE WHAT IT IS A stff bar that rests on a support called a fulcrum A slantng surface connectng a lower level to a hgher level A wheel wth a rod, called an ael, through ts center: both parts move together HOW IT HELPS US WORK Lfts or moves loads Thngs move up or down t Lfts or moves loads EXAMPLES Shovel, nutcracker, seesaw, crow-bar, elbow Slde, stars, ramp, escalator Doorknob, pencl sharpener, bke Quck Overvew of Each SIMPLE MACHINES PULLEY WEDGE SCREW WHAT IT IS A grooved wheel wth a rope or cable around t A type of nclned plane wth a sharp edge. The wedge moves, the nclned plane stays stll. An nclned plane wrapped around a cylnder. Works wth a lever. HOW IT HELPS US WORK Moves thngs up, down, or across Pushes thngs apart Rases weghts, presses or fastens objects EXAMPLES Curtan rod, tow truck, mn-blnd, flag pole, crane Ae blade Screws, nuts Lever LEVER: The lever s a smple machne made wth a bar free to move about a fed pont called a fulcrum. There are three types of levers. A frst class lever s lke a teeter-totter or see-saw. One end wll lft an object (chld) up just as far as the other end s pushed down. A second class lever s lke a wheel barrow. The long handles of a wheel barrow are really the long arms of a lever. A thrd class lever s lke a fshng pole. When the pole s gven a tug, one end stays stll but the other end flps n the ar catchng the fsh. Inclned Plane INCLINED PLANE : An nclned plane s a smple machne wth no movng parts. It s smply a straght slanted surface. For eample: a ramp Wheel & Ale WHEEL AND AXLE : A wheel and ale s a modfcaton of a pulley. A wheel s fed to a shaft. The wheel and shaft must move together to be a smple machne. Sometmes the wheel has a crank or handle on t. Eamples of wheel and ales nclude roller skates and doorknobs. 13
Pulley PULLEY: A pulley s a smple machne made wth a rope, belt or chan wrapped around a grooved wheel. A pulley works two ways. It can change the drecton of a force or t can change the amount of force. A fed pulley changes the drecton of the appled force. ( E. Rasng the flag ). A movable pulley s attached to the object you are movng. Wedge WEDGE: A wedge s a modfcaton of an nclned plane that moves. It s made of two nclned planes put together. Instead of the resstance beng moved up an nclned plane, the nclned plane moves the resstance. Screw SCREW : A screw s a smple machne that s lke an nclned plane. It s an nclned plane that wraps around a shaft. (Smple) Machnes A machne can never output more work (energy) than s put nto t. At best, Work out = Work n Work n Machne Work out Mechancal Advantage Machnes can t multply work or energy, but they can multply force. Mechancal advantage measures how much a machne multples force. MA = Force machne eerts Force you eert Effcency The effcency of a machne tells how much of the energy (work) that goes nto the machne actually does useful work. It s usually epressed as a percent. Effcency = Energy output Energy nput 100% 14