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1 EWTO S LAWS OF MOTIO ewton 1 st lw o Lw of Ineti Evey body continues to be in its stte of est o of unifom motion until nd unless nd until it is compelled by n extenl foce to chnge its stte of est o of unifom motion. Ineti The popety by vitue of which body opposes ny chnge in its stte of est o of unifom motion is known s ineti. Gete the mss of the body gete is the ineti. Tht is mss is the mesue of the ineti of the body. umeicl Appliction If, F = 0 ; u = constnt Physicl Appliction 1. When moving bus suddenly stops, pssenge s hed gets jeked in the fowd diection.. When sttioney bus suddenly stts moving pssenge s hed gets jeked in the bckwd diection. 3. On hitting used mttess by stick, dust pticles come out of it. 4. In ode to ctch moving bus sfely we must un fowd in the diection of motion of bus. 5. Wheneve it is equied to jump off moving bus, we must lwys un fo shot distnce fte jumping on od to pevent us fom flling in the fowd diection. Key Concept In the bsence of extenl pplied foce velocity of body emins unchnged. ewton nd lw Rte of chnge of momentum is diectly popotionl to the pplied foce nd this chnge lwys tkes plce in the diection of the pplied foce. dp F dt 8

2 dp = F (hee popotionlity constnt is 1) dt putting, p = mv F = dp dt F = dmv dt F = mdv + vdm dt dt F = mdv (if m is constnt dm/dt = 0) dt F = m ote :- Above esult is not ewton s second lw the it is the conditionl esult obtined fom it, unde the condition when m = constnt. umeicl Appliction = F et m Whee F et is the vecto esultnt of ll the foces cting on the body. F 1 F F 6 m F 3 m F et F 5 F 4 Whee, F et = F 1 + F + F 3 + F 4 + F 5 + F 6 9

3 Hoizontl Plne Physicl Appliction i) Cse - 1 Body kept on hoizontl plne is t est. Fo veticl diection = (since body is t est) ii) Body kept on hoizontl plne is cceleting hoizontlly unde single hoizontl foce. Fo veticl diection = (since body is t est) Fo hoizontl diection F = m iii) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde two hoizontl foces. (F 1 > F ) Fo veticl diection = (since body is t est) F F 1 Fo hoizontl diection F 1 - F = m iv) Body kept on hoizontl plne is cceleting hoizontlly unde single inclined foce FSin F Fo veticl diection + FSin = (since body is t est) FCos Fo hoizontl diection FCos = m v) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde n inclined foce nd hoizontl foce. F 1 Sin F 1 Fo veticl diection + F 1 Sin = (since body is t est) F F 1 Cos F Fo hoizontl diection F 1 Cos F = m 30

4 vi) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde two inclined foces cting on opposite sides. Fo veticl diection F 1Sin F 1 + F 1 Sin = + F SinФ (since body is t est) F CosФ Fo hoizontl diection F 1 Cos F CosФ = m F F SinФ Ф F 1Cos Inclined Plne i) Cse - 1 Body sliding feely on inclined plne. Pependicul to the plne = Cos (since body is t est) Sin Pllel to the plne Sin = m ii) Cse - Body pulled pllel to the inclined plne. Pependicul to the plne = Cos (since body is t est) Pllel to the plne F - Sin = m Sin Cos Cos F iii) Cse - 3 Body pulled pllel to the inclined plne but cceleting downwds. Pependicul to the plne = Cos (since body is t est) Pllel to the plne Sin Sin - F = m F Cos 31

5 iv) Cse - 4 Body cceleting up the incline unde the effect of two foces cting pllel to the incline. F 1 Pependicul to the plne = Cos (since body is t est) F Pllel to the plne Sin F 1 - F - Sin = m v) Cse - 5 Body cceleting up the incline unde the effect of hoizontl foce. Pependicul to the plne = Cos + F 1 Sin (since body is t est) Pllel to the plne F 1 Cos - Sin = m Sin Cos F 1Cos F 1 F 1Sin Cos vi) Cse - 6 Body cceleting down the incline unde the effect of hoizontl foce nd gvity. FSin Pependicul to the plne + FSin = Cos (since body is t est) F FCos Pllel to the plne Sin FCos + Sin = m Cos vii) Cse - 7 Body cceleting up the incline unde the effect of two hoizontl foces cting on opposite sides of body nd gvity. F Cos F 1Sin F Pependicul to the plne F 1 + F 1 Sin = Cos + F Sin(since body is t est) FSin F 1Cos Pllel to the plne Sin F Cos - F 1 Cos - Sin = m Cos 3

6 Veticl Plne i) Cse - 1 Body pushed ginst the veticl plne by hoizontl foce nd moving veticlly downwd. Fo hoizontl diection = m (since body is t est) Fo veticl diection F = F ii) Cse - Body pushed ginst the veticl plne by hoizontl foce nd pulled veticlly upwd. F Fo veticl diection F - = m Fo hoizontl diection (since body is t est) F 1 = F 1 iii) Cse - 3 Body pushed ginst the veticl plne by inclined foce nd cceletes veticlly upwd. FCos Fo hoizontl diection = FSin (since body is t est) Fo veticl diection FCos = m F FSin iv) Cse - 3 Body pushed ginst the veticl plne by inclined foce nd cceletes veticlly downwd. Fo hoizontl diection FSin = FSin (since body is t est) Fo veticl diection FCos + = m F FCos 33

7 Tension In A Light Sting Foce pplied by ny line object such s sting, ope, chin, od etc. is known s it s tension. Since sting is highly flexible object so it cn only pull the object nd cn neve push. Hence tension of the sting lwys cts wy fom the body to which it is ttched iespective of the diection. Tension of the sting, being of pulling ntue, lwys cts wy fom the body to which it is ttched Physicl Appliction i) Flexible wie holding the lmp pulls the lmp in upwd diection nd pulls the point of suspension in the downwd diection. ii) Rope holding the bucket in the well pulls the bucket in the upwd diection nd the pulley in the downwd diection. iii) Rope ttched between the cttle nd the peg pulls the cttle towds the peg nd peg towds the cttle. iv) When block is pulled by the chin, the chin pulls the block in fowd diection nd the peson holding the chin in evese diection. Key Point In cse of light sting, ope, chin, od etc. tension is sme ll long thei lengths. T 1 P T Conside point P on light (mssless) sting. Let tensions on eithe side of it be T 1 nd T espectively nd the sting be cceleting towds left unde these foces. Then fo point P T 1 - T = m Since sting is consideed to be light mss m of point P is zeo T 1 - T = 0 T 1 = T 34

8 i) Cse - 1 Two bodies connected by sting e plced on smooth hoizontl plne nd pulled by hoizontl foce. 1 m T T m 1 F m g m 1 g Fo veticl equilibium of m 1 nd m 1 = m 1 g nd = m g Fo hoizontl cceletion of m 1 nd m F T = m 1 nd T = m (Since both the bodies e connected to the sme single sting they hve sme cceletion) ii) Cse - Two bodies connected by hoizontl sting e plced on smooth hoizontl plne nd pulled by inclined foce. 1 FSin F m T T m 1 FCos m g m 1 g Fo veticl equilibium of m 1 nd m 1 + FSin = m 1 g nd = m g Fo hoizontl cceletion of m 1 nd m FCos T = m 1 nd T = m (since both the bodies e connected to the sme single sting they hve sme cceletions) iii) Cse - 3 Two bodies connected by inclined sting e plced on smooth hoizontl plne nd pulled by inclined foce. 35

9 1 FSin F TCos m TSin T m 1 T TSin FCos TCos m g m 1 g Fo veticl equilibium of m 1 nd m 1 + FSin = m 1 g + TSin nd + TSin = m g Fo hoizontl cceletion of m 1 nd m FCos TCos = m 1 nd TCos = m (since both the bodies e connected to the sme single sting they hve sme cceletions) iv) Cse - 4 Two bodies connected by sting mde to ccelete up the incline by pplying foce pllel to the incline. 1 F m gsin m g T m gcos m 1gSin T m 1g m 1gCos Fo equilibium of m 1 nd m in the diection pependicul to the plne 1 = m 1 gcos nd = m gcos Fo cceletion of m 1 nd m up the incline F - T - m 1 gsin = m 1 nd T - m gsin = m Tension of A light Rigid Rod Foce pplied by od is lso known s its tension. Since od is igid, it cnnot bend like sting. Hence od cn pull s well s push. Tension of od cn be of pulling s well s pushing ntue but one t time. Tension of od ttched to the body my be diected towds s well s wy fom the body. T T FFF T T Tension of od is pulling both the blocks Tension of od is pushing both the blocks 36

10 Physicl Appliction i) Pills suppoting the house pushes the house in the upwd diection nd pushes the gound in the downwd diection. ii) Wooden bs used in the chi pushes the gound in the downwd diection nd pushes the seting top in the upwd diection. iii) Pllel bs ttched to the ice-cem tolley pushes the tolley in the fowd diection nd pushes the ice-cem vendo in the bckwd diection.(when the tolley is being pushed by the vendo) iv) Rod holding the ceiling fn pulls the fn in the upwd diection nd pulls the hook ttched to the ceiling in the downwd diection. v) Pllel ods ttched between the ct nd the bull pulls the ct in the fowd diection nd pulls the bull in the bckwd diection. Diffeent Cses of Light Rigid Rod i) Cse - 1 Rod ttched fom the ceiling nd suppoting the block ttched to its lowe end. Since the block is t est T T = T ii) Cse - Rod is ttched between two blocks plced on the hoizontl plne nd the blocks e cceleted by pushing foce. 1 Fo veticl equilibium of m 1 nd m m 1 T T m 1 = m 1 g nd = m g F m Fo hoizontl cceletion of m 1 nd m F T = m 1 nd T = m m 1 g m g (Since both the bodies connected to the od will hve sme cceletion) iii) Cse - 3 Rod is ttched between two blocks plced on the hoizontl plne nd the blocks e cceleted by pulling foce. 1 m T T m 1 F Fo veticl equilibium of m 1 nd m 1 = m 1 g nd = m g m g m 1 g 37

11 Fo hoizontl cceletion of m 1 nd m F T = m 1 nd T = m (Since both the bodies e connected to the sme od they hve sme cceletion) iv) Cse - 4 Rod is ttched between two blocks plced on the incline plne nd the blocks e cceleted by pushing pllel to the incline. Fo veticl equilibium of m 1 nd m m gsin 1 T 1 = m 1 gcos nd = m gcos T m gcos m g Fo cceletion of m 1 nd m pllel to F m1gsin the incline m 1gCos F m 1 gsin - T = m 1, m 1g T m gsin = m Fixed Pulley It is simple mchine in the fom of cicul disc o im suppoted by spokes hving goove t its peiphey. It is fee to otte bout n xis pssing though its cente nd pependicul to its plne. Key Point In cse of light pulley, tension in the ope on both the sides of the pulley is sme (to be poved in the ottionl mechnics) T 1 T Anticlockwise Toque - Clockwise Toque = Moment of Ineti x Angul cceletion T 1 x - T x = Iα Since the pulley is light nd hence consideed to be mssless, it s moment of ineti I = 0 T 1 x - T x = 0 T 1 x = T x T 1 = T 38

12 Diffeent Cses of Fixed Pulley i) Cse - 1 Two bodies of diffeent msses (m 1 > m ) e ttched t T 1 two ends of light sting pssing ove smooth light pulley Fo veticl equilibium of pulley T 1 T 1 = T + T = T Fo veticl cceletion of m 1 nd m m 1 g - T = m 1 nd T - m g = m T T m 1 cceletes downwds nd m cceletes upwds(m 1 >m ) m 1 m ii) Cse - Two bodies of diffeent msses e ttched t two ends of light sting pssing ove light pulley. m 1 is plced on hoizontl sufce nd m is hnging feely in i. Fo veticl equilibium m 1 = m 1 g m 1 T T T m 1 g m g T Fo hoizontl cceletion of m 1 T = m 1 m 1 g T Fo veticlly downwd cceletion of m m g - T = m T m g iii) Cse - 3 Two bodies of diffeent msses e ttched t two ends of light sting pssing ove light pulley. m 1 is plced on n inclined sufce nd m is hnging feely in i. 39

13 Fo equilibium of m 1 pependicul to incline plne = m 1 gcos Fo cceletion of m 1 up the incline plne T T - m 1 gsin = m 1 m 1gSin m 1 m Fo veticlly downwd cceletion of m m g - T = m m 1g m 1gCos m g T T Key Point Movble Pulley The pulley which moves in itself is known s movble pulley. In cse of light movble pulley, cceletion of body (pulley) goes on decesing on incesing the numbe of stings ttched to it. Tht is the body ttched with two opes moves with hlf the cceletion of the body ttched with single ope. Length of the sting is constnt z x + y + z = L (Constnt) Diffeentiting both sides with espect to t (Time) dx + dy + dz = dl dt dt dt dt y v 1 + v + 0 = 0 (z nd L e constnt) x v 1 + v = 0 Agin diffeentiting both sides with espect to t dv 1 + dv = 0 1 dt dt m 1 m 1 + = 0 1 = - Tht is cceletion of m 1 (body ttched to single sting) is opposite nd twice the cceletion of m (body ttched to double sting) Diffeent Cses of Light Movble Pulley i) Cse - 1 Mss m 1 is ttched t one end of the sting nd the othe end is fixed to igid suppot. Mss m is ttched to the light movble pulley. 40

14 w T 1 T Fo veticl cceletion of m 1 T 1 m 1 g - T = m 1 (m 1 is connected to single sting) Fo veticl cceletion of m T T T m g = m (m 1 cceletes downwds nd m cceletes upwds since m 1 >m ) T T T Fo the clmp holding the fist pulley T 1 = T Fo the clmp holding the movble pulley T m 1 m T - T = m pulley T - T = 0 (light pulley) T = T m 1g m g ii) Cse - Mss m 1 is ttched t one end of the sting nd plced on smooth hoizontl sufce nd the othe end is fixed to igid suppot fte pssing though light movble suspended pulley. Mss m is ttched to the light movble pulley. Fo veticl equilibium of m 1 = m 1 g m 1 T T T Fo hoizontl cceletion of m 1 T = m 1 m 1g T Fo veticl motion of m T T m g T = m iii) Cse - 3 Mss m 1 is ttched to the movble pulley nd plced on smooth hoizontl sufce. One end of the sting is ttched to the clmp holding the pulley fixed to the hoizontl sufce nd fom its othe end mss m suspended. Fo veticl equilibium of m 1 T T T m m g = m 1 g m 1 Fo hoizontl motion of m 1 T T = m 1 m 1g T Fo veticl motion of m m m g - T = m T T m g 41

15 iv) Cse - 4 Mss m 1 is ttched to movble pulley nd plced on smooth inclined sufce. Mss m is is suspended feely fom fixed light pulley. t T T Fo equilibium of m 1 pependicul to incline plne T = m 1 gcos T x T T Fo cceletion of m 1 up the incline plne T m T - m 1 gsin = m 1 m 1 Fo veticlly downwd cceletion of m m g - T = m m 1gSin m 1g m 1gCos ewton 3 d lw o Lw of Action nd Rection Evey ction is opposed by n equl nd opposite ection. o Fo evey ction thee is n equl nd opposite ection. F 1 m 1 F 1 F 1 is the foce on the fist body (m 1 ) due to second body (m ) F 1 is the foce on the second body (m ) due to fist body (m 1 ) m If F 1 is ction then F 1 ection nd if F 1 is ction then F 1 ection umeicl Appliction Foce on the fist body due to second body (F 1 ) is equl nd opposite to the foce on the second body due to fist body (F 1 ). m g F 1 = - F 1 Physicl Appliction i) When we push ny block in the fowd diection then block pushes us in the bckwd diection with n equl nd opposite foce. ii) Hose pulls the od ttched to the ct in the fowd diection nd the tension of the od pulls the ct in the bckwd diection. 4

16 iii) Eth pulls the body on its sufce in veticlly downwd diection nd the body pulls the eth with the sme foce in veticlly upwd diection. iv) While wlking we push the gound in the bckwd diection using sttic fictionl foce nd the gound pushes us in the fowd diection using sttic fictionl foce. v) When peson sitting on the hose whips the hose nd hose suddenly cceletes, the sddle on the bck of the hose pushes the peson in the fowd diection using sttic fictionl foce nd the peson pushes the sddle in the bckwd diection using sttic fictionl foce. ote oml ection of the hoizontl sufce on the body is not the ection of the weight of the body becuse weight of the body is the foce with which eth ttcts the body towds its cente, hence its ection must be the foce with which body ttcts eth towds it. Line Momentum It is defined s the quntity of motion contined in the body. Mthemticlly it is given by the poduct of mss nd velocity. It is vecto quntity epesented by p. p = mv Pinciple Of Consevtion Of Line Momentum It sttes tht in the bsence of ny extenl pplied foce totl momentum of system emins conseved. Poof- We know tht, F = m F = mdv dt F = dmv dt if, F = 0 F = dp dt dp = 0 dt p = Constnt (diffeentition of constnt is zeo) p initil = p finl 43

17 Physicl Appliction i) Recoil of gun when bullet is fied in the fowd diection gun ecoils in the bckwd diection. ii) When peson jumps on the bot fom the shoe of ive, bot long with the peson on it moves in the fowd diection. iii) When peson on the bot jumps fowd on the shoe of ive, bot stts moving in the bckwd diection. iv) In ocket populsion fuel is ejected out in the downwd diection due to which ocket is popelled up in veticlly upwd diection. Diffeent Cses of Consevtion of Line Momentum Recoil of gun Let mss of gun be m g nd tht of bullet be m b. Initilly both e t est, hence thei initil momentum is zeo. p i = m g u g + m b u b = 0 Finlly when bullet ushes out with velocity v g, gun ecoils with velocity v b, hence thei finl momentum is p f = m g v g + m b v b Since thee is no extenl pplied foce, fom the pincipl of consevtion of line momentum p f = p f m g v g + m b v b = 0 m g v g = -m b v b v g = - m b v b m g Fom bove expession it must be cle tht 1. Gun ecoils opposite to the diection of motion of bullet.. Gete is the mss of mullet m b o velocity of bullet v b gete is the ecoil of the gun. 3. Gete is the mss of gun m g, smlle is the ecoil of gun. 44

18 Impulse nd Impulsive Foce Impulsive Foce The foce which cts on body fo vey shot dution of time but is still cpble of chnging the position, velocity nd diection of motion of the body up to lge extent is known s impulsive foce. Exmple - 1. Foce pplied by foot on hitting footbll.. Foce pplied by boxe on punching bg. 3. Foce pplied by bt on bll in hitting it to the boundy. 4. Foce pplied by moving tuck on dum. ote- Although impulsive foce cts on body fo vey shot dution of time yet its mgnitude vies pidly duing tht smll dution. Impulse Impulse eceived by the body duing n impct is defined s the poduct of vege impulsive foce nd the shot time dution fo which it cts. I = F vg x t Reltion Between Impulse nd Line Momentum Conside body being cted upon by n impulsive foce, this foce chnges its mgnitude pidly with the time. At ny instnt if impulsive foce is F then elementy impulse impted to the body in the elementy time dt is given by di = F x dt Hence totl impulse impted to the body fom time t 1 to t is t I = Fdt But fom ewton s second lw we know tht F = dp dt Fdt = dp t 1 Theefoe, p I = dp p 1 p I = [p] p 1 I = p p 1 Hence impulse impted to the body is equl to the chnge in its momentum. 45

19 Gph Between Impulsive Foce nd Time With the time on x xis nd impulsive foce on y xis the gph of the following ntue is obtined F t 1 t t Ae enclosed unde the impulsive foce nd time gph fom t 1 to t gives the impulse impted to the body fom time t 1 to t. Physicl Appliction i) While ctching bll plye lowes his hnd to sve himself fom getting hut. ii) Vehicles e povided with the shock bsobes to void jeks. iii) Buffes e povided between the bogies of the tin to void jeks. iv) A peson flling on cemented floo eceive moe jek s comped to tht flling on sndy floo. v) Glss wes e wpped in stw o ppe befoe pcking. Equilibium of Concuent Foces If the numbe of foces ct t the sme point, they e clled concuent foces. The condition o the given body to be in equilibium unde the numbe of foces cting on the body is tht these foces should poduce zeo esultnt. The esultnt of the concuent foces cting on body will be zeo if they cn be epesented completely by the sides of closed polygon tken in ode. F 1 + F + F 3 + F 4 + F 5 = 0 F 3 F 4 F F 4 F 1 F 5 F 3 F F 1 F 5 46

20 Lmi s Theoem It sttes tht the thee foces cting t point e in equilibium if ech foce is popotionl the sine of the ngle between the othe two foces. F 1 F β β α F 1 ϒ F 3 ϒ α F F 3 F 1 = F = F 3 Sin α Sin β Sin ϒ Inetil nd on-inetil Fme of Refeence Fme of efeence is ny fme with espect to which the body is nlyzed. All the fmes which e t est o moving with constnt velocity e sid to be inetil fme of efeence. In such fme of efeence ll the thee lws of ewton e pplicble. Any cceleted fme of efeence is sid to be non-inetil fme of efeence. In such fmes ll the thee lws of ewton e not pplicble s such. In ode to pply ewton s lws of motion in non-inetil fme, long with ll othe foces pseudo foce F = m must lso be pplied on the body opposite to the diection of cceletion of the fme. T T T TCos T TCos TSin TSin m 47

21 Inetil Fme of Refeence (Fme outside the cceleted c) Fo veticl equilibium of body TCos = Fo hoizontl cceletion of body, s the body is cceleted long with the c when obseved fom the extenl fme TSin = m =0 Theefoe, Tn = /g Inetil Fme of Refeence (Fme ttched to the cceleted c) Fo veticl equilibium of body TCos = Fo hoizontl equilibium of the body, s the body is t est when obseved fom the fme ttched to the c TSin = m Theefoe, Tn = /g Since body is t est when obseved fom the non-inetil fme ttched to the cceleted c pseudo foce F = m is pplied on the body opposite to the cceletion of the c which blnce the hoizontl component of tension of the sting TSin cting on the body. ote- Fom which eve fme we my obseve the sitution, finl esult lwys comes out to be the sme. Reding of Sping Blnce Reding of sping blnce is equl to the tension in the sping of the blnce but mesued in kilogm. Reding of Weighing Mchine Reding = T kgf g Reding of weighing mchine is equl to the noml ection pplied by the mchine but mesued in kilogm. Reding = kgf g LIFT T T T T T T =0 48

22 Obseve Outside the Lift Lift Acceleting Veticlly Up Moving up with incesing velocity. o Moving down with decesing velocity. Fo veticl motion of body T - = m T = + m T = m(g + ) Lift Acceleting Veticlly Up Moving up with constnt velocity. o Moving down with constnt velocity. Fo veticl motion of body =0 T = Lift Acceleting Veticlly Down Moving up with decesing velocity. o Moving down with incesing velocity. Fo veticl motion of body - T = m T = - m T = m(g - ) T T T T T T ' =0 Lift Acceleting Veticlly Up Moving up with incesing velocity. o Moving down with decesing velocity. Since body is t est T= but, T = m(g + ) theefoe, g = g + Whee g is ppent cceletion due to gvity inside the lift. Obseve Inside the Lift (Body is t est ccoding to the obseve inside the lift) Lift Acceleting Veticlly Up Moving up with constnt velocity. o Moving down with constnt velocity. Since body is t est T = but, T = theefoe, g = g Whee g is ppent cceletion due to gvity inside the lift. Lift Acceleting Veticlly Down Moving up with decesing velocity. o Moving down with incesing velocity. Since body is t est T = But, T = m(g - ) theefoe, g = g - Whee g is ppent cceletion due to gvity inside the lift. 49

23 MEMORY MAP ewton s 1 st Lw If F = 0 u = Constnt ewton s nd Lw F = dp/dt = F et /m Pinciple of Consevtion of Momentum If, F ext = 0; p i = p f ewton s Lws of Motion FORCE ewton s 3 d Lw F 1 = F 1 F 1 = - F 1 Impulse I = F AVG t I = p 50

24 FRICTIO Fiction - The popety by vitue of which the eltive motion between two sufces in contct is opposed is known s fiction. Fictionl Foces - Tngentil foces developed between the two sufces in contct, so s to oppose thei eltive motion e known s fictionl foces o commonly fiction. Types of Fictionl Foces - Fictionl foces e of thee types :- 1. Sttic fictionl foce. Kinetic fictionl foce 3. Rolling fictionl foce Sttic Fictionl Foce - Fictionl foce cting between the two sufces in contct which e eltively t est, so s to oppose thei eltive motion, when they tend to move eltively unde the effect of ny extenl foce is known s sttic fictionl foce. Sttic fictionl foce is self djusting foce nd its vlue lies between its minimum vlue up to its mximum vlue. Minimum vlue of sttic fictionl foce - Minimum vlue of sttic fictionl foce is zeo in the condition when the bodies e eltively t est nd no extenl foce is cting to move them eltively. f s(min) = 0 Mximum vlue of sttic fictionl foce - Mximum vlue of sttic fictionl foce is µ s (whee µ s is the coefficient of sttic fiction fo the given pi of sufce nd is the noml ection cting between the two sufces in contct) in the condition when the bodies e just bout to move eltively unde the effect of extenl pplied foce. f s(mx) = µ s Theefoe, f s(min) f s f s(mx) 0 f s µ s Kinetic Fictionl Foce - Fictionl foce cting between the two sufces in contct which e moving eltively, so s to oppose thei eltive motion, is known s kinetic fictionl foce. It s mgnitude is lmost constnt nd is equl to µ k whee µ k is the coefficient of kinetic fiction fo the given pi of sufce nd is the noml ection cting between the two sufces in contct. It is lwys less thn mximum vlue of sttic fictionl foce. Since, Theefoe, f k = µ k f k < f s(mx) = µ s µ k < µ s µ k < µ s 51

25 Limiting Fictionl Foce The mximum vlue of sttic fictionl foce is the mximum fictionl foce which cn ct between the two sufces in contct nd hence it is lso known s limiting fictionl foce. Lws of Limiting Fictionl Foce 1. Sttic fiction depends upon the ntue of the sufces in contct.. It comes into ction only when ny extenl foce is pplied to move the two bodies eltively, with thei sufces in contct. 3. Sttic fiction opposes the impending motion. 4. It is self djusting foce. 5. The limiting fictionl foce is independent of the e of contct between the two sufces. Cuse of Fiction Old View - The sufces which ppe to be smooth s seen though ou nked eyes e ctully ough t the micoscopic level. Duing contct, the pojections of one sufce penette into the depessions of othe nd vice ves. Due to which the two sufces in contct fom sw tooth joint opposing thei eltive motion. When extenl foce is pplied so s to move them eltively this joint opposes thei eltive motion. As we go on incesing the extenl pplied foce the opposition of sw tooth joint lso goes on incesing up to the mximum vlue known s limiting fictionl foce (µ s ) fte which the joint suddenly beks nd the sufces stt moving eltively. Afte this the opposition offeed by the sw tooth joint slightly deceses nd comes to est t lmost constnt vlue (µ k ) Moden View Accoding to moden theoy the cuse of fiction is the tomic nd molecul foces of ttction between the two sufces t thei ctul point of contct. When ny body comes in contct with ny othe body then due to thei oughness t the micoscopic level they come in ctul contct t sevel points. At these points the toms nd molecules come vey close to ech othe nd intemolecul foce of ttction stt cting between them which opposes thei eltive motion. Contct Foce - The foces cting between the two bodies due to the mutul contct of thei sufces e known s contct foces. The esultnt of ll the contct foces cting between the bodies is known s esultnt contct foce. Exmple 5

26 fiction (f) nd noml ection () e contct foces nd thei esultnt (F c ) is the esultnt is the esultnt contct foce. F c F f F c = f + Since mximum vlue of fictionl foce is Limiting fictionl foce (µ s ) Theefoe mximum vlue of contct foce is F c(mx) = (µ s ) + F c(mx) = µ s + 1 F c(mx) = µ s + 1 Angle of Fiction The ngle between the esultnt contct foce (of noml ection nd fiction) nd the noml ection is known s the ngle of fiction. Tn = f F c = Tn -1 f mx = Tn -1 mx = Tn -1 mx = Tn -1 f mx µ s f µ s Angle of Repose The ngle of the inclined plne t which body plced on it just begins to slide is known s ngle of epose. Pependicul to the plne = Cos (since body is t est) f s F Pllel to the plne when body is t est Sin Sin = f s Cos When body is just bout to slide 53

27 Sin = f s(mx) = µ s = µ s Cos Tn = µ s = Tn -1 µ s ote - Angle of epose is equl to the mximum vlue of ngle of fiction Rolling Fictionl Foce - Fictionl foce which opposes the olling of bodies (like cylinde, sphee, ing etc.) ove ny sufce is clled olling fictionl foce. Rolling fictionl foce cting between ny olling body nd the sufce is lmost constnt nd is given by µ. Whee µ is coefficient of olling fiction nd is the noml ection between the olling body nd the sufce. f = µ ote Rolling fictionl foce is much smlle thn mximum vlue of sttic nd kinetic fictionl foce. f << f k < f s(mx) µ << µ k < µ s µ << µ k < µ s Cuse of Rolling Fiction When ny body olls ove ny sufce it cuses little depession nd smll hump is ceted just hed of it. The hump offes esistnce to the motion of the olling body, this esistnce is olling fictionl foce. Due to this eson only, hd sufces like cemented floo offes less esistnce s comped to soft sndy floo becuse hump ceted on hd floo is much smlle s comped to the soft floo. f v(diection of olling) eed to Convet Kinetic Fiction into Rolling Fiction Of ll the fictionl foces olling fictionl foce is minimum. Hence in ode to void the we nd te of mchiney it is equied to convet kinetic fictionl foce into olling fictionl foce nd fo this eson we mke the use of bll-beings. Rings hving goove on its inne side Rings hving goove on its oute side Steel bll tpped between the goves 54

28 Fiction: A ecessy Evil Although fictionl foce is non-consevtive foce nd cuses lots of wstge of enegy in the fom of het yet it is vey useful to us in mny wys. Tht is why it is consideed s necessy evil. Advntges of Fiction - i) Fiction is necessy in wlking. Without fiction it would hve been impossible fo us to wlk. ii) Fiction is necessy fo the movement of vehicles on the od. It is the sttic fictionl foce which mkes the cceletion nd etdtion of vehicles possible on the od. iii) Fiction is helpful in tying knots in the opes nd stings. iv) We e ble to hold nything with ou hnds by the help of fiction only. Disdvntges of Fiction - i) Fiction cuses we nd te in the mchiney pts. ii) Kinetic fiction wstes enegy in the fom of het, light nd sound. iii) A pt of fuel enegy is consumed in ovecoming the fiction opeting within the vious pts of mchiney. Methods to Reduce Fiction i) By polishing Polishing mkes the sufce smooth by filling the spce between the depessions nd pojections pesent in the sufce of the bodies t micoscopic level nd thee by educes fiction. ii) By pope selection of mteil Since fiction depends upon the ntue of mteil used hence it cn be lgely educed by pope selection of mteils. iii) By lubicting When oil o gese is plced between the two sufces in contct, it pevents the sufce fom coming in ctul contct with ech othe. This convets solid fiction into liquid fiction which is vey smll. Hoizontl Plne Physicl Appliction i) Body kept on hoizontl plne is t est nd no foce is pplied. Fo veticl equilibium = f fiction = 0 (fiction is opposing foce nd thee is no extenl pplied foce) ii) Body kept on hoizontl plne is t est unde single hoizontl foce. Fo veticl equilibium = (since body is t est) Fo hoizontl equilibium (since body is t est) F = f s f s F 55

29 iii) Body kept on hoizontl plne is just bout to move. Fo veticl diection = (since body is t est) Fo hoizontl diection (since body is just bout to move) F = f s = f s(mx) = µ s f s = f s(mx) = µ s iv) Body kept on hoizontl plne is cceleting hoizontlly. F Fo veticl diection = (since body is t est) Fo hoizontl diection F f k = m F µ k = m f k = µ k v) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde single upwd inclined foce. Fo veticl diection + FSin = (since body is t est) FSin Fo hoizontl diection FCos FCos - f k = m f k = µ k FCos - µ k = m vi) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde single downwd inclined foce. Fo veticl diection = FSin + (since body is t est) FCos Fo hoizontl diection FCos - f k = m f k = µ k FSin FCos - µ k = m F F F vii) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde n inclined foce nd n opposing hoizontlly pplied foce. Fo veticl diection + FSin = (since body is t est) Fo hoizontl diection FCos - F 1 - f k = m FCos - F 1 - µ k = m 56 f k = µ k FSin F 1 F FCos

30 vi) Body kept on hoizontl plne is cceleting hoizontlly towds ight unde two inclined foces cting on opposite sides. F 1Sin F 1 Fo veticl diection(since body is t est) + F 1 Sin = + F SinФ F CosФ Ф Fo hoizontl diection F F SinФ F 1Cos F 1 Cos F CosФ - µ k = m f k = µ k Inclined Plne i) Cse - 1 Body is t est on inclined plne. Pependicul to the plne = Cos (since body is t est) Pllel to the plne (since body is t est) Sin = f s ii) Cse - Body is just bout to move on inclined plne. Pependicul to the plne = Cos (since body is t est) Pllel to the plne (since body is t est) Sin = f s = f s(mx) = µ s Sin Sin f s Cos f s = f s(mx) = µ s Cos iii) Cse - 3 Body is cceleting downwds on inclined plne. Pependicul to the plne = Cos (since body is t est) Pllel to the plne Sin - f k = m Sin - µ k = m Sin Cos f k 57

31 iv) Cse - 4 Body is cceleting up the incline unde the effect of foce cting pllel to the incline. Pependicul to the plne = Cos (since body is t est) Pllel to the plne Sin F - f k - Sin = m F - µ k - Sin = m f k Cos F v) Cse - 5 Body cceleting up the incline unde the effect of hoizontl foce. Pependicul to the plne = Cos + FSin (since body is t est) FCos Pllel to the plne Sin FCos - Sin - f k = m f k Cos FCos - Sin - µ k m Veticl Plne i) Cse - 1 Body pushed ginst the veticl plne by hoizontl foce nd is t est. f s Fo hoizontl diection (since body is t est) F = Fo veticl diection = f s FSin F F ii) Cse - Body pushed ginst the veticl plne by hoizontl foce nd pulled veticlly upwd F 1 Fo hoizontl diection (since body is t est) F = F Fo veticl diection F 1 - f s = m f s 58

32 iii) Cse - 3 Body pushed ginst the veticl plne by inclined foce nd cceletes veticlly upwd. FCos F Fo hoizontl diection = FSin (since body is t est) Fo veticl diection FCos - - f s = m FSin f s 59

33 MEMORY MAP Sttic Fictionl Foce 0 f s µ s Kinetic Fictionl Foce f k = µ k Angle of Fiction = Tn -1 f/ 0 Tn -1 µ s A ECESSARY EVIL FRICTIO µ << µ k < µ s Rolling Fictionl Foce f = µ Angle of Repose = Tn -1 µ s 60

34 CIRCULAR MOTIO Cicul Motion When body moves such tht it lwys emins t fixed distnce fom fixed point then its motion is sid to be cicul motion. The fixed distnce is clled the dius of the cicul pth nd the fixed point is clled the cente of the cicul pth. Unifom Cicul Motion Cicul motion pefomed with constnt speed is known s unifom cicul motion. Angul Displcement Angle swept by the dius vecto of pticle moving on cicul pth is known s ngul displcement of the pticle. Exmple : ngul displcement of the pticle fom P 1 to P is. Reltion Between Angul Displcement nd Line Displcement Since, Angle = c dius Anglul Displcement = c P 1 P dius = s Angul Velocity Rte of chnge of ngul displcement of body with espect to time is known s ngul displcement. It is epesented by ω. P P 1 Avege Angul Velocity It is defined s the tio of totl ngul displcement to totl time tken. ω vg = Totl Angul Displcement Totl Time Tken ω vg = t Instntneous Angul Velocity Angul velocity of body t some pticul instnt of time is known s instntneous ngul velocity. O 61

35 Avege ngul velocity evluted fo vey shot dution of time is known s instntneous ngul velocity. ω = Lim ω vg = t 0 t ω = d dt Reltion Between Angul Velocity nd Line Velocity We know tht ngul velocity Putting, = s/ ω = d dt ω = d (s/) dt ω = 1 ds dt ω = v v = ω Time Peiod of Unifom Cicul Motion Totl time tken by the pticle pefoming unifom cicul motion to complete one full cicul pth is known s time peiod. In one time peiod totl ngle otted by the pticle is nd time peiod is T. Hence ngul velocity ω = T T = ω Fequency - umbe of evolutions mde by the pticle moving on cicul pth in one second is known s fequency. f = 1 = ω T Centipetl Acceletion When body pefoms unifom cicul motion its speed emins constnt but velocity continuously chnges due to chnge of diection. Hence body is continuously cceleted nd the cceletion expeienced by the body is known s centipetl cceletion (tht is the cceletion diected towds the cente). 6

36 v P v 1 v v O s C B R v v 1 v v P 1 A Conside pticle pefoming unifom cicul motion with speed v. When the pticle chnges its position fom P 1 to P its velocity chnges fom v 1 to v due to chnge of diection. The chnge in velocity fom P 1 to P is v which is diected towds the cente of the cicul pth ccoding to tingle lw of subtction of vectos. Fom figue OP 1 P nd ABC e simil, hence pplying the condition of simility BC = P 1 P 1 AB O P 1 v = s v v = v s Dividing both sides by t, v = v s t t Tking limit t 0 both sides, Lim v = v Lim t 0 t t 0 t dv = vds dt dt = v Putting v = ω, = ω Since the chnge of velocity is diected towds the cente of the cicul pth, the cceletion esponsible fo the chnge in velocity is lso diected towds cente of cicul pth nd hence it is known s centipetl cceletion. Centipetl Foce Foce esponsible fo poducing centipetl cceletion is known s centipetl foce. Since centipetl cceletion is diected towds the cente of the cicul pth the centipetl foce is lso diected towds the cente of the cicul pth. If body is pefoming unifom cicul motion with speed v nd ngul velocity ω on cicul pth of dius, then centipetl cceletion is given by 63

37 F c F c = mv = mω et Acceletion of Body Pefoming on-unifom Cicul Motion When body pefoms non-unifom cicul motion its speed i.e. mgnitude of instntneous velocity vies with time due to which it expeiences tngentil cceletion T long with the centipetl cceletion C. Since both the cceletions ct simultneously on body nd e mutully pependicul to ech othe, the esultnt cceletion R is given by thei vecto sum. C R R = T + C R = T + C Physicl Appliction of Centipetl Foce i) Cse - 1 Cicul motion of stone tied to sting. Centipetl foce is povided by the tension of the sting F c = mv = T ii) Cse - Cicul motion of electon ound the nucleus. Centipetl foce is povided by the electosttic foce of ttction between the positively chged nucleus nd negtively chged electon T F c = mv = F E 64

38 iii) Cse - 3 Cicul motion of plnets ound sun o stellites ound plnet. Centipetl foce is povided by the gvittionl foce of ttction between the plnet nd sun F c = mv = F g iv) Cse - 4 Cicul motion of vehicles on hoizontl od. Centipetl foce is povided by the sttic fictionl foce between the od nd the tye of the vehicle. F c = mv = f s v) Cse - 5 Cicul motion of block on otting pltfom. Centipetl foce is povided by the sttic fictionl foce between the block nd the pltfom. F c = mv = f s vi) Cse - 6 Cicul motion of mud pticles sticking to the wheels of the vehicle. Centipetl foce is povided by the dhesive foce of ttction between the mud pticles nd the tyes of the vehicle. F c = mv = F dhesive At vey high speed when dhesive foce is unble to povide necessy centipetl foce, the mud pticles fly off tngentilly. In ode to pevent the pticles fom stining ou clothes, mud-guds e povided ove the wheels of vehicle. v v 65

39 vii) Cse - 7 Cicul motion of tin on hoizontl tck. Centipetl foce is povided by the hoizontl component of the ection foce pplied by the oute tck on the inne pojection of the oute wheels H F c = mv = Hoizontl H viii) Cse - 8 Cicul motion of toy hnging fom ceiling of vehicle. C moving with constnt velocity on hoizontl od TSin T TCos C tking tun with constnt velocity on hoizontl od Wheneve c tkes tun, sting holding the toy gets tilted outwd such tht the veticl component of the tension of sting blnces the weight of the body nd the hoizontl component of tension povides the necessy centipetl foce. TSin = mv TCos = Theefoe, Tn = v ix) Cse - 9 Conicl pendulum. g T TSin T TCos Wheneve bob of pendulum moves on hoizontl cicle it s sting genetes cone. Such pendulum is known s conicl pendulum. The veticl component of the tension of the sting blnces the weight of the body nd the hoizontl component of tension povides the necessy centipetl foce. 66

40 TSin = mv TCos = Theefoe, Tn = v x) Cse - 10 Well of deth. g Cos Cos In the well of deth, the ide ties to push the wll due to its tngentil velocity in the outwd diection due to which wll pplies noml ection in the inwd diection. The veticl component of the noml ection blnces the weight of the body nd its hoizontl component povides the necessy centipetl foce. Sin = mv Cos = Theefoe, Tn = v xi) Cse - 11 Tuning of eo plne. F P F P Cos F P g F P Sin While tking tun eo-plne tilts slightly inwds due to which it s pessue foce lso gets tilted inwds due to which it s pessue foce lso gets tilted inwds such tht it s veticl component blnces the weight of the body nd the hoizontl component povides the necessy centipetl foce. 67

41 F P Sin = mv F P Cos = Theefoe, Tn = v xi) Cse - 11 Bnking of Rods In cse of hoizontl od necessy centipetl foce mv / is povided by sttic fictionl foce. When hevy vehicles move with high speed on shp tun (smll dius) then ll the fctos contibute to huge centipetl foce which if povided by the sttic fictionl foce my esult in the ftl ccident. To pevent this ods e bnked by lifting thei oute edge. Due to this, noml ection of od on the vehicle gets tilted inwds such tht it s veticl component blnces the weight of the body nd the hoizontl component povides the necessy centipetl foce. g n ncos c nsin nsin = mv ncos = 68

42 Theefoe, Tn = v g xii) Cse - 1 Bending of Cyclist In cse of cyclist moving on hoizontl cicul tck necessy centipetl foce is povided by sttic fictionl foce cting pllel long the bse. As this fictionl foce is not pssing fom the cente of mss of the system it tends to otte the cycle long with the cyclist nd mke it fll outwd of the cente of the cicul pth. To pevent himself fom flling, the cyclist lens the cycle inwds towds the cente of the cicle due to which the noml ection of the sufce of od on the cycle lso lens inwd such tht tht its veticl component blnces the weight of the body nd the hoizontl component povides the necessy centipetl foce. Sin Cos Sin = mv Cos = Theefoe, Tn = v g 69

43 xiii) Cse - 13 Motion of Bll in Bowl ω A o Cos Cos When the bowl ottes with some ngul velocity ω. The veticl component of the noml ection of the bowl on the bll blnces the weight of the body nd its hoizontl component povides the necessy centipetl foce. Sin = mv Cos = Theefoe, Tn = v xiv) Cse - 14 Motion of tin on the bnked tcks. At the tuns tcks e bnked by slightly elevting the oute tcks with espect to the inne ones. This slightly tilts the tin inwds towds the cente of the cicul pth due to which the noml ection of the tcks on the tin lso gets slightly tilted inwds such tht the veticl component of the noml ection blnces the weight of the tin nd it s hoizontl component povides the necessy centipetl foce. g Cos 70

44 Sin = mv Cos = Theefoe, Tn = v g Veticl Cicul Motion Wheneve the plne of cicul pth of body is veticl its motion is sid to be veticl cicul motion. Veticl Cicul Motion of Body Tied to Sting v A T A T A A Conside body of mss m tied to sting nd pefoming veticl cicul motion on cicul pth of dius. At the topmost point A of the body weight of the body nd tension T A both e cting in the veticlly downwd diection towds the cente of the cicul pth nd they togethe povide centipetl foce. T A + = mv A Citicl velocity t the top most point As we go on decesing the v A, tension T A lso goes on decesing nd in the citicl condition when v A is minimum tension T A = 0. The minimum vlue of v A in this cse is known s citicl velocity T A(Citicl) t the point A. Fom bove 0 + = mv A(Citicl) v A(Citicl) = g v A(Citicl) = g 71

45 If the velocity t point A is less thn this citicl velocity then the sting will slg nd the body in spite of moving on cicul pth will tend to fll unde gvity. Citicl velocity t the lowe most point A T B V B T B B Tking B s efeence level of gvittionl potentil enegy nd pplying enegy consevtion E A = E B P A + K A = P B + K B + 1mv A = 0 + 1mv B Putting, v A = g + 1m( g) = 0 + 1mv B 4 + = mv B 5 = mv B v B = 5g This is the minimum possible velocity t the lowe most point fo veticl cicul motion known s citicl velocity t point B. v B(Citicl) = 5g Tension t lowemost point in citicl condition Fo lowemost point B net foce towds the cente is centipetl foce. Tension T B cts towds the cente of the cicul pth whees weight cts wy fom it. Hence, T B = mv B Putting, v B = 5g 7

46 T B = m5g T B = 6 Hence in citicl condition of veticl cicul motion of body tied to sting velocities t topmost nd lowemost be (g) nd (5g) espectively nd tensions in the stings be 0 nd 6 espectively. Genel Condition fo Slgging of Sting in Veticl Cicul Motion Fo the body pefoming veticl cicul motion tied to sting, slgging of sting occus in the uppe hlf of the veticl cicle. If t ny instnt sting mkes ngle with veticl then pplying net foce towds cente is equl to centipetl foce, we hve Fo slgging T = 0, Cos T v T + Cos = mv 0 + Cos = mv v = gcos Cse-1 At Topmost point = 0, theefoe v = g Cse- At = 60 o, theefoe v = gcos60 = g/ Cse-3 When sting becomes hoizontl tht is t = 90 o, v = gcos90 = 0 Velocity Reltionship t diffeent Points of Veticl Cicul Motion Let initil nd finl velocities of the body pefoming veticl cicul motion be v 1 nd v nd the ngle mde by sting with the veticl be 1 nd. Tking lowemost point of veticl cicul pth s efeence level nd pplying enegy consevtion, 73

47 V 1 V Cos 1 Cos 1 E 1 = E P 1 + K 1 = P + K ( + Cos 1 ) + 1mv 1 = ( + Cos ) + 1mv (Cos 1 Cos ) = 1m(v v 1 ) (v v 1 ) = g(cos 1 Cos ) Veticl Cicul Motion of Body Attched to Rod Since od cn neve slg hence in the citicl sitution body ttched to the od my ech the topmost position A of the veticl cicul pth with lmost zeo velocity. In this cse its weight cts in veticlly downwd diection nd tension of od cts on the body in the veticlly upwd diection. Applying net foce towds cente is equl to centipetl foce, A v A T A T A - T A = mv A Putting v A = 0 (fo citicl condition) - T A = 0 T A = 74

48 Citicl velocity nd Tension t the lowe most point A T B T B V B B Tking B s efeence level of gvittionl potentil enegy nd pplying enegy consevtion E A = E B P A + K A = P B + K B + 1mv A = 0 + 1mv B Putting, v A = 0(fo citicl condition) + 0 = 0 + 1mv B 4 = mv B v B = 4g This is the minimum possible velocity t the lowe most point fo veticl cicul motion known s citicl velocity t point B. v B(Citicl) = 4g Tension t lowemost point in citicl condition Fo lowemost point B pplying net foce towds cente is equl to centipetl foce. Tension T B cts towds the cente of the cicul pth whees weight cts wy fom it in veticlly downwd diection. Hence, Putting, v B = 4g T B = mv B T B = m4g T B = 5 75

49 Hence in citicl condition of veticl cicul motion of body ttched to the od velocities t topmost nd lowemost be 0 nd 4g espectively nd tensions in the od be (pushing ntue) nd 5 (pulling ntue) espectively. Motion of A Body Ove Spheicl Sufce V Sin Cos A body of mss m is moving ove the sufce of the smooth sphee of dius. At ny instnt when the dius of sphee pssing though the body mkes ngle with the veticl the tngentil velocity of the body is v. Since net foce towds the cente is centipetl foce we hve Cos = mv = Cos mv if v inceses deceses nd when the body just loses contct with the sphee = 0. Putting = 0, 0 = Cos mv mv = Cos v = g Cos This is the minimum velocity t which the body loses contct nd it is the mximum velocity t which the body emins in contct with the sufce. 76

50 CETRIFUGAL FORCE It is pseudo foce expeienced by body which is pt of the cicul motion. It is non-elistic foce nd comes into ction only when the body is in cicul motion. Once the cicul motion of the body stops, this foce ceses to ct. Its mgnitude is exctly sme s tht of centipetl foce but it cts opposite to the diection of the centipetl foce tht is in the dilly outwd diection. Fme of efeence ttched to body moving on cicul pth is non-inetil fme since it n cceleted fme. So when eve ny body is obseved fom this fme pseudo foce F = m = mv / = mω must be pplied on the body opposite to the diection of cceletion long with the othe foces. Since the cceletion of the fme in cicul motion is centipetl cceletion = v / diected towds the cente of the cicul pth, the pseudo foce pplied on the bodies obseved fom this fme is F = mv / diected wy fom the cente of the cicul pth. This pseudo foce is temed s centifugl foce. F CF F Centifugl = mv = mω (Diected in dilly outwd diection) CETRIFUGE It is n pptus used to septe cem fom milk. It woks on the pincipl of centifugl foce. It is cylindicl vessel otting with high ngul velocity bout its centl xis. When this vessel contins milk nd ottes with high ngul velocity ll the pticles of milk stt moving with the sme ngul velocity nd stt expeiencing centifugl foce F Centifugl = mω in dilly outwd diection. Since centifugl foce is diectly popotionl to the mss of the pticles, mssive pticles of milk on expeiencing gete centifugl foce stts depositing on the oute edge of the vessel nd lighte cem pticles on expeiencing smlle centifugl foce e collected ne the xis fom whee they e septed pt. ω Cem 77

51 MEMORY MAP ω = v = = f T Centipetl Foce F C = mv = mω Rdilly Outwd Diection Body Tied to Sting V top = (g) nd V bottom = (5g) T top = 0 nd T bottom = 6 Cicul Motion Citicl Condition Fo Veticl Cicul Motion Centifugl Foce F CF = mv = mω Rdilly Inwd Diection Body Attched to Rod V top = 0 nd V bottom = (4g) T top = - nd T bottom = 5 Citicl Condition of Veticl Cicul MOtion 78

52 Vey Shot Answe Type 1 Mk Questions 1. Is net foce needed to keep body moving with unifom velocity?. Is ewton s nd lw (F = m) lwys vlid. Give n exmple in suppot of you nswe? 3. Action nd ection foces do not blnce ech othe. Why? 4. Cn body emin in stte of est if moe thn one foce is cting upon it? 5. Is the centipetl foce cting on body pefoming unifom cicul motion lwys constnt? 6. The sting is holding the mximum possible weight tht it could withstnd. Wht will hppen to the sting if the body suspended by it stts moving on hoizontl cicul pth nd the sting stts geneting cone? 7. Wht is the ection foce of the weight of book plced on the tble? 8. Wht is the mximum cceletion of vehicle on the hoizontl od? Given tht coefficient of sttic fiction between the od nd the tyes of the vehicle is µ. 9. Why guns e povided with the shoulde suppot? 10. While pddling bicycle wht e the types of fiction cting on e wheels nd in which diection? Answe 1. o.. It is vlid in n inetil fme of efeence. In non-inetil fme of efeence (such s c moving long cicul pth), ewton s nd lw doesn t hold ppently. 3. Since they e cting on diffeent bodies. 4. Yes, if ll the foces cting on it e in equilibium. 5. o, only its mgnitude emins constnt but its diection continuously goes on chnging. 6. It will bek becuse tension in the sting inceses s soon s the body stts moving. 7. The foce with which the book ttcts the eth towds it. 8. mx = fs(mx)/m = µ/m = µ/m = µg. 9. So tht the ecoil of gun my be educed by poviding suppot to the gun by the shouldes. 10. Sttic fiction in fowd diection nd olling fiction in bckwd diection. Shot Answe Type Mks Questions 1. Explin why the wte doesn t fll even t the top of the cicle when the bucket full of wte is upside down otting in veticl cicle?. The displcement of pticle of mss 1kg is descibed by s = t + 3t. Find the foce cting on pticle? (F = 6) 3. A pticle of mss 0.3 kg is subjected to foce of F = -kx with k = 15 m 1. Wht will be its initil cceletion if it is elesed fom point 10 cm wy fom the oigin? ( = - 5 ms ) 4. Thee foces F 1, F nd F 3 e cting on the pticle of mss m which is sttiony. If F 1 is emoved, wht will be the cceletion of pticle? ( = F 1 /m) 79

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