Chapter 12: Velocity, acceleration, and forces

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1 To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable bu ery small effec oer shor imes or disances, while he effecs of he moion of he earh around he sun can safely be ignored. A mass is said o be a res, saionary, or no moing, only if is posiion wih respec o he earh remains he same oer a ime ineral. A mass is said o be in moion, or is moing if is posiion changes oer a ime ineral. The measuremen of he posiion of he objec a a single ime canno deermine if he objec is a res or moing. For an objec o be a res beween wo imes, he posiion mus be he same a all inermediae imes. Newon s s law saes ha under he acion of balanced forces, a mass will remain a res or moe wih a consan speed and direcion. Newon s 3 rd law, describing he exisance of an acion reacion pair of forces a a poin of conac beween wo masses, is rue in all saes of moion: a res, moing wih consan speed and direcion, or moing wih a changing speed or direcion. B. Characerisics of moion Kineic energy and momenum are measures of he moion of a mass and use in heir definiions he insananeous elociy,, wih a speed,, and direcion (+ or ). Eeryone is familiar wih speed, and knows ha a consan speed can be measured by aking he raeled disance (a posiie scalar) and diiding by he ime of rael, = s. The elociy is obained by adding a sign in fron of he speed,, following he same conenion for direcion as used for displacemen. Formally, for elociies ha could be changing, he aerage elociy is defined as: s =, (.) where he displacemen ecor, s, is he same as he one used in he definiion of work ( w = F s). A consan elociy is no ery ineresing; he ne exernal force acing on he mass is zero. When he elociy changes oer ime, howeer, we are obsering he effecs of a non zero ne exernal force. Measuring he alue of he aerage elociy, as gien by Equaion., oer a long ime,, aerages oer eeryhing ha happened o he objec, and does no gie a good picure of wha effec he force is haing on he objec a any specific ime. The moion a any specific ime (or posiion), is called he insananeous elociy,, and can be deermined using Equaion. in he limi of ery small. A saemen ha = a a single ime (or posiion), howeer, gies no eidence for, or 5

2 To Feel a Force Chaper Spring, agains, he acion of a ne exernal force. Measuremens of local changes in deermine he presence of a ne exernal force on an objec. C. Effecs of an ne exernal force. The inernal ension or compression forces on he masses a he ends of an ideal spring, he graiaional forces beween wo masses, and he fricional forces beween masses in conac, are examples of acion - reacion force pairs. The definiions work, kineic and poenial energy, momenum and he conseraion laws ha goern heir behaior were obained by considering he acion of he inernal force pairs of an ideal spring on objecs wih mass. When one mass was small and he oher was ery large (he mass of he earh) he predicable effecs of he elasic force on he large mass allowed us o infer a law for he conseraion of energy. The effecs of inernal forces on finie masses a boh ends of a spring allowed us o infer a law for he conseraion of momenum. A force is called an exernal force if he oher member of he acion - reacion force pair is no considered. Considered in isolaion, he graiaional force on a small mass m near he surface of he earh is an exernal force wih a consan alue F G = mg (is weigh). A force of his magniude also acs on he earh in he opposie direcion, bu he effecs of his small force on he moion of he earh can safely be ignored. This allows us o use he earh as a poin of reference for he moion of he small mass. D. Newon s nd law of moion: effecs of a consan exernal force on a mass. Newon discoered ha here were wo properies of moion affeced by a ne exernal force: kineic energy, and momenum. An exernal force changes he kineic energy of a mass raeling a disance, and an exernal force changes he momenum of a mass raeling for a period of ime. The momenum change is equal o he produc of he aerage force and he acion ime, p= p + F, where momenum, p= m. Therefore, he relaionship beween he aerage force, he mass, and he change in elociy is F = p p = m F = m ( ) ( ) The obsered change in he elociy of a mass, diided by he ime of he moion, is called he aerage acceleraion, a =. If he ne force acing on a mass doesn change, hen F = F, and a = a, resuling in he mos famous of Newon s discoeries, F. = m a. (.) 6

3 To Feel a Force Chaper Spring, If here is more han one exernal force acing on a mass hen is acceleraion will be proporional o he ne force obained as usual by a ecor sum. E. Acceleraion due o graiy near he surface of he earh The graiaional force on a mass m near he surface of he earh is F G = mg (+ is upward) and is consan. If he only force acing on a mass is he graiaional force near he surface of he earh, Newon s second law, Equaion. aboe, yields: F = FG ma = mg and wih cancellaion of he common facor of m one finds ha he acceleraion of a mass near he surface of he earh is a = g. (.3) This is a saemen ha he acceleraion of a mass near he surface of he earh is always downward and always has he magniude, g = 98. m/s, wih he unis of an acceleraion. The graiaional force ecor on a mass hrown upward near he surface of he earh is F G = mg. Throughou he enire free fligh, he acceleraion ecor is a = g, and herefore, he elociy is always changing ( is no zero). A he ime he mass reaches is highes poin, where he speed, =, he mass is sill moing: a any ime earlier or laer he speed is no zero, and he heigh of he mass will be lower han is maximum heigh. F. The equaions of moion for a mass under a consan acceleraion The definiion of acceleraion if consan, a =, and of aerage elociy, = + s, were discussed aboe. They are he firs wo equaions of moion: = + a, (.4) and s= +. (.5) The kineic energy change is equal o he work done on he mass by all forces (including conseraie and non conseraie), m ( ) = F s. Replacing F by ma and diiding boh sides by m/, yields he hird equaion of moion: = a s. (.6) Eliminaing he final elociy,, from Equaion.6, yields he fourh equaion of moion: s= + a (.7) 7

4 To Feel a Force Chaper Spring, The subsiuion, a = g, will yield he four equaions of moion for a mass in free fligh near he surface of he earh: = + ( g ) s= + = ( g) s s= + ( g) If here is no ne force acing on he mass, he acceleraion will be zero, a =, he elociy will no change, =, and he four equaions reduce o he one for a consan elociy: s =. An aserisk (*) in he able idenifies four of he fie moion ariables,,, a, s, and, used in each of he equaions. In soling a moion problem, here will always be a leas hree equaions ha are poenially useful, because hey conain he unknown moion ariable. For problems where he iniial elociy,, and anoher ariable of ineres are no gien, he iniial elociy can be deermined from he equaion aboe missing he ariable of ineres. These relaionships beween,, a, s, and, hae energy conseraion hidden wihin hem and here is much ha hese equaions can explain abou he moion of an objec. They are quie useful in siuaions where he applied forces or he acceleraion is known. They gie ery lile informaion, howeer, on how acceleraed moion will be perceied if he acceleraed objec is a human being. The nex chaper addresses his deficiency. Table. Moion equaions for a consan acceleraion Eqs..8. Equaion a s missing Free Fligh Eq. = + a s = + ( g ) s= + a s = + = a s = ( g) s s= + a s = + ( g) 8

5 To Feel a Force Chaper Spring, Chaper Summary In he saes, a res, saionary, and no moing, an objec remains a he same place for a gien period of ime. An objec is MOVING (i is no a res, and i is no saionary), if i has a zero speed a one ime, bu a non-zero speed any shor ime earlier or laer. If an objec is subjeced o a consan acceleraion and he elociy changes sign, here is a cerain ime and locaion where he elociy is zero. A his ime he objec is no a res and no saionary, i is moing. I is moing because i does no say a he posiion where he elociy is zero for een he shores period of ime. Newon s nd law of moion: a consan ne exernal force, F, he ecor sum of all forces acing on a mass m, will cause a consan acceleraion gien by F = m a. The acceleraion of an objec in free fall near he surface of he earh is he consan, a = g. The four moion equaions for a consan acceleraion are shown in Table.. 9

1. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to t = T. a

1. The graph below shows the variation with time t of the acceleration a of an object from t = 0 to t = T. a Kinemaics Paper 1 1. The graph below shows he ariaion wih ime of he acceleraion a of an objec from = o = T. a T The shaded area under he graph represens change in A. displacemen. B. elociy. C. momenum.