FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com CIRCULAR MOTION When a paticle moves in a plane such that its distance fom a fixed (o moving) point emains constant then its motion is called as the cicula motion with espect to that fixed (o moving) point. That fixed point is called cente and the distance is called adius. KINEMATICS OF CIRCULAR CULAR MOTION Vaiables of Motion (a) Angula Position The angle made by the position vecto with given line (efeence line) is called angula position. Cicula motion is a two dimensional motion of motion in a plane. Suppose a paticle P is moving in a cicle of adius and cente O. The position of the paticle P at a given instant may be descibed by the angle θ between OP and OX. This angle θ is called the angula position of the paticle. As the paticle moves on the cicle its angula position θ change. Suppose the point otates an angle θ in time t. (b) Angula Displacement (c) Definition: Angle otated by a position vecto of the moving paticle with some efeence line is called angula displacement. Impotant points:. It is dimensionless and has pope unit (SI unit) adian while othe units ae degee o evolution π ad 360 ev. Infinitely small angula displacement is a vecto quantity but finite angula displacement is not because the addition of the small angula displacement is commutative while fo lage is not. dθ + dθ dθ + θ but θ + θ θ + θ d 3. Diection of small angula displacement is decided by ight hand thumb ule. When the figues ae diected along the motion of the point then thumb will epesents the diection of angula displacement. 4. Angula displacement can be diffeent fo diffeent obseves (i) (ii) Angula Velocity ω Aveage Angula Velocity Total Angle of Rotation Total time taken ω av ; ω av θ θ t t θ t whee θ and θ ae angula position of the paticle at time t and t Instantaneous Angula Velocity The ate at which the position vecto of a paticle w..t. the cente otates, is called as instantaneous angula velocity with espect to the cente. θ ω lim t 0 t Impotant points: CIRCULAR MOTION d θ d t Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com. It is an axial vecto with dimensions [T - ] and SI unit ad/s. FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Ex. Ans.. Fo a igid body as all points will otate though same angle in same time, angula velocity is a chaacteistic of the body as a whole, e.g., angula velocity of all points of eath about its own axis is (π/4) ad/h. 3. If a body makes n otations in t seconds then angula velocity in adian pe second will be 4. If ω av πn t If T is the peiod and f the fequency of unifom cicula motion dθ θ a bt + ct then ω b + ct π ω av π f T Is the angula velocity of otation of hou hand of a watch geate of smalle than the angula velocity of Eath s otation about its own axis. Houhand completes one otation in hous while Eath completes one otation in 4 hous. So, angula velocity of hou hand is double the angula velocity of Eath. π ω. T (d) Angula Acceleation α (i) Aveage Angula Acceleation : Let ω and ω be the instantaneous angula speeds at times t and t espectively, then the aveage angula acceleation α av is defined as ω ω ω αav t t t (ii) Instantaneous Angula Acceleation : It is the limit of aveage angula acceleation as t appoaches zeo, i.e., Impotant points: ω dω α lim ω t 0 t dω dθ. It is also an axial vecto with dimension [T - ] and unit ad/s.. If α 0, cicula motion is said to be unifom. dθ dω d θ 3. As ω, α, i.e., second deivative of angula displacement w..t. time gives angula acceleation. RELATION BET WEEN SPEED AND ANGULAR VELOCIT OCITY ω lim t θ θ t dθ The ate of change of angula velocity is called the angula acceleation (α). Thus, dω d θ α The linea distance PP tavelled by the paticle in time t is o lim t 0 s θ s lim t t 0 θ t Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com o o s t v ω dθ Hee, v is the linea speed of the paticle. Diffeentiating again with espect to time, we have dω a t o at α Hee, a t is the ate of change of speed (not the ate of change of velocity). This is called the tangential acceleation of the paticle. Late, we will see that a t is the component of net acceleation a of the paticle moving in a cicle along the tangent. Ex. A paticle tavels in a cicle of adius 0 cm at a speed that unifom inceases. If the speed changes fom 5.0 m/s to 6.0 m/s in.0s, find the angula acceleation. Sol. The tangential acceleation is given by a 6. 0 5. 0 m/s. 0 0.5 m/s. The angula acceleation is α a t / 0. 5m / s 0cm.5 ad/s. RADIAL AND TANGENTIAL T ACCELERAA CCELERATION Unit vectos along the adius and the tangent v t Conside a paticle P moving in a cicle of adius and cente at oigin O. The angula position of the paticle at some instant is say θ. Let us hee define two unit vectos, one is ê (called adial unit vecto) which is along OP and the othe is ê t (called the tangential unit vecto) which is pependicula to OP. Now, since ê ê t We can wite these two vectos as and ê cos θ î + sin θ ĵ ê t sin θ î + cos θ ĵ v t Velocity and acceleation of paticle in cicula motion : The position vecto of paticle P at the instant shown in figue can be witten as OP ê o (cos θ î + sin θ ĵ ) The velocity of the paticle can be obtained by diffeentiating with espect to time t. Thus, v d ( sin θ î + cos θ ĵ ) ω a Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 3 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Ex. 3 Sol. Ex. 4 d dω ω ( sinθî + cosθĵ) + ( sinθî + cosθĵ) dω ω [cos θ î + sin θ ĵ ] + a ω ê + ê t ê t Thus, acceleation of a paticle moving in a cicle has two components one is along ê t (along tangent) and the othe along ê (o towads cente). Of these the fist one is called the tangential acceleation. (a t ) and the othe is called adial o centipetal acceleation (a ). Thus. and a t ate of change of speed v a w v Hee, the two components ae mutually pependicula. Theefoe, net acceleation of the paticle will be : t a a + a ( ω ) + v + Following thee points ae impotant egading the above discussion:. In unifom cicula motion, speed (v) of the paticle is constant, i.e., 0. Thus, a t 0 and a a ω. In acceleated cicula motion, positive, i.e., at is along paticle is paallel to velocity v because v ω ê t and a ê t o tangential acceleation of 3. In deceleated cicula motion, negative and hence, tangential acceleation is anti-paallel to velocity v. A paticle moves in a cicle of adius.0 cm at a speed given by v 4t, whee v is in cm/s and t in seconds. (a) Find the tangential acceleation at t s. (b) Find total acceleation at t s. (a) Tangential acceleation o a t d a t (4t) 4 cm/s a C R v (4) 8 a a + a (4) + (8) 4 5 m/s ê t t C A boy whils a stone in a hoizontal cicle of adius.5 m and at height.0 m above level gound. The sting beaks, and the stone files off hoizontally and stikes the gound afte taveling a hoizontal distance of 0 m. Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 4 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com What is the magnitude of the cetipetal acceleation of the stone while in cicula motion? FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com h Sol. t 0.64s g 9.8 Ex. 5 Sol. Ex. 6 Sol. Ex. 7 Sol. v 0 5.63 m/s t a R v B 0.45 m/s Find the magnitude of the linea acceleation of a paticle moving in a cicle of adius 0 cm with unifom speed completing the cicle in 4s. The distance coveed in completing the cicle is π π 0 cm. The linea speed is v π /t The linea acceleation is a π 0 cm 4s v 5 π cm/s. ( 5πcm / s) 0cm.5 π cm/s. A paticle moves in a cicle of adius 0 am. Its linea speed is given by v t whee t is in second and v in mete/second. Find the adical and tangential acceleation at t 3s. The linea speed at t 3s is v t 6 m/s. The adical acceleation at t 3s is a v / The tangent acceleation is 36m / s 0. 0m d ( ) a t m/s t. 80 m/s. Two paticles A and B stat at the oigin O and tavel in opposite diections along the cicula path at constant speeds v A 0.7 m/s and v B.5 m/s, espectively. Detemine the time when they collide and the magnitude of the acceleation of B just befoe this happens..5t + 0.7t πr 0π t 0π 4.3 s. a R v B 0.45 m/s RELATIVE ANGULAR VELOCIT OCITY Angula velocity is defined with espect to the point fom which the position vecto of the moving paticle is dawn Hee angula velocity of the paticle w..t. O and A will be diffeent ω PO Definition: dα ; ω PO dβ A β α O P P Ref. lin. Relative angula velocity of a paticle A with espect to the othe moving paticle B is the angula velocity of the position vecto of A with espect to B. That means it is the ate at which position vecto of A with espect to B otates at that instant Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 5 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com ( V ) AB ω AB AB (V AB ) VA sinθ + VB sinθ AB w AB Impotant points: vb sinθ + vb sinθ Relative velocity of A w..t. B pependicula to line AB Sepeation between A and B. If two paticles ae moving on the same cicle o diffeent coplana concentic cicles in same diection with diffeent unifom angula speed ω A and ω B espectively, the angula velocity of B elative to A fo an obseve at the cente will be ω BA ω B ω A dθ So the time taken by one to complete one evolution aound O w..t. the othe π T ω el π ω ω TT T T. If two paticles ae moving on two diffeent concentic cicles with diffeent velocities then angula velocity of B elative to A as obseved by A will depend on thei positions and velocities. conside the case when A and B ae closet to each othe moving in same diection as shown in figue. In this situation v v v v v so, el B A B A el ω BA B (v el A el ) B v B B A v A A (v el ) Relative velocity pependicula to position vecto RELATIONS AMONG ANGULAR VARIABLES These elations ae also efeed as equations of otational motion and ae ω ω 0 + αt - () θ ω 0 t + αt - () ω ω 0 + αθ - (3) v o a T O ds a d θ, ω o α These ae valid only if angula acceleation is constant and ae analogous to equations of tanslatoy motion, i.e., Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 6 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com v u + at; s ut + (/) at and v u + as FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com RADIUS OF CURVATURE Any cuved path can be assume to be made of infinite cicula acs. Radius of cuvatue at a point is the adius of the cicula ac at a paticula point which fits the cuve at that point. F c mv R R mv F C mv F Foce pependicula to velocity (centipetal foce) If the equation of tajectoy of a paticle is given we can find the adius of cuvatue of the instantaneous cicle by using the fomula, R dy ( ) 3/ + dx d y dx DYNAMICS OF CIRCULAR MOTION. F In cicula motion o motion along any cuved path Newton s law is applied in two pependicula diections one along the tangent and othe pependicula to it. i.e. towads cente. The component of net foce along the cente is called centipetal foce. The component of net foce along the tangent is called tangential foce. tangential foce (F t ) Ma t M M α centipetal foce (F c ) m ω mv 8. A small block of mass 00 g moves with unifom speed in a hoizontal cicula goove, with vetical side walls, of adius 5 cm. If the block takes.0s to complete one ound, find the nomal contact foce by the slide wall of the goove. Sol. The speed of the block is v π ( 5cm). 0s The acceleation of the block is 0.785 m/s v ( 0. 785m / s) a.5 m/s. 0. 5 towads the cente. The only foce in this diection is the nomal contact foce due to the slide walls. Thus fom Newton s second law, this foce is ma (0.00 kg) (.5 m/s ) 0.5 N CENTRIPETAL FORCE Concepts : This is necessay esultant foce towads the cente called the centipetal foce. (i) (ii) (iii) F mv mω A body moving with constant speed in a cicle is not in equilibium. It should be emembeed that in the absence of the centipetal foce the body will move in a staight line with constant speed. It is not a new kind of foce which acts on bodies. In fact, any foce which is diected towads the cente may povide the necessay centipetal foce. Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 7 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com A small block of mass m, is at est elative to tuntable which otates with constant angula speed ω. FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com CIRCULAR CULAR MOTION IN HORIZONTAL PLANE Ex. 9 A ball of mass m attached to a light and inextensible sting otates in a hoizontal cicle of adius with an angula speed ω about the vetical. If we daw the foce diagam of the ball. We can easily see that the component of tension foce along the cente gives the centipetal foce and component of tension along vetical balances the gavitation foce. An aicaft executes a hoizontal loop of adius km with a steady speed of 900 km h. Compae its centipetal acceleation with the acceleation due to gavity. Sol. km 0 3 m; v 900 km h 900 8 5 ms 50 m s Centipetal acceleation, Now, v 50 50 a c 3 m s 0 6.5 m s a c g 6.5 0 6.5 Ex. 0 A stone tied to the end of a sting 80 cm long is whiled in a hoizontal cicle with a constant speed. If the stone makes 4 evolutions in 5 s. What is the magnitude and diection of acceleation of the stone? Sol. 80 cm 0.80 m; ω Centipetal acceleation of ω 4 evolutions 5s π 4 4 4 0.8 9.9 ms 7 7 5 5 At evey point, the acceleation is along the adius and towads the cente. Ex. A paticle of mass m is suspended fom a ceiling though a sting of length L. The paticle moves in a hoizontal cicle of adius. Find (a) the speed of the paticle and (b) the tension in the sting. Such a system Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 8 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com Sol. is called a conical pendulum. The situation is shown in figue. The angle θ made by the sting with the vetical is given by sin θ / L...(i) The foces on the paticle ae (a) the tension T along the sting and (b) the weight mg vetically downwad. The paticle is moving in a cicle with a constant speed v. Thus, the adical acceleation towads the cente has magnitude v /. Resolving the foces along the adial diection and applying. Newton s second law, Tsin θ m(v / )...(ii) As thee is no acceleation in vetical diections, we have fom Newton s law, Tcos θ mg...(iii) Dividing (ii) by (iii), v tan θ g o, v gtan θ And fom (iii), T mg cosθ g Using (i), v ( L ) 4 and mgl ( L ) Ex. Two blocks each of mass M ae connected to the ends of a light fame as shown in figue. The fame is otated about the vetical line of symmety. The od bakes if the tension in it exceeds T 0. Find the maximum fequency with which the fame may be otated without baking the od. Sol.. Conside one of the blocks. If the fequency of evolution is f, the angula velocity is ϖ π f. The acceleation towads the cente is v / l ϖ l 4 π f l. The only hoizontal foce on the block is the tension of the od. At the point of baking, this foce is T 0. So fom Newton s law, o, f T 0 M.4 π f l T0 π M l / Q. A paticle of mass 4 g attached to a sting of 70 cm length is whiled ound in a hoizontal cicle. If the peiod of evolution is second, calculate the tension. Ans. 9680 dyne Q. A sting beaks unde a load of 50 kg. A mass of kg is attached to one end of the sting 0 m long and is otated in hoizontal cicle. Calculate the geatest numbe of evolutions that the mass can make without beaking the sting. Ans. n.4 evolutions pe second. MOTION IN A VERTICAL CIRCLE To undestand this conside the motion of a small body (say stone) tied to a sting and whiled in a vetical cicle. If at any time the body is at angula position θ, as shown in the figue, the foces acting on it ae tension T in the sting along the adius towads the cente and the weight of the body mg acting vetically down wads. Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 9 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com Applying Newton s law towads cente FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com mv mv T mg cosθ o T + mg cosθ The body will move on the cicula path only and only if T min > 0 (as if T min 0, the sting will slack and the body will fall down instead of moving on the cicle). So fo completing the cicle, i.e., looping the loop mv H mg 0 i.e., v g...() H Now applying consevation of mechanical enegy between highest point H and lowest point L we get v L 5g i.e., fo looping the loop, velocity at lowest point must be 5g. In case of motion in a vetical plane tension is maximum at lowest position and in case of looping the loop T min 6mg. CONDITION FOR F OSCILLATION OR LEAVING THE CIRCLE CLE In case of non unifom cicula motion in a vetical plane if velocity of body at lowest point is lesse than 5 g, the paticle will not complete the cicle in vetical plane. Now it can eithe oscillate about the lowest point o afte eaching a cetain height may loose contact with the path. Fom the theoy of looping the loop we know that if v L of a body at lowest point is such that g < vl < 5g 5g, the body will loop the loop. So if the velocity the body will move along the cicle fo θ > 90 and will not each upto highest point but will leave the cicle somewhee between 90 < θ < 80. Hee it is woth noting that at the point of leaving the cicle T 0 but v 0 0. This all is shown in figue. Ex. 3 A simple pendulum is constucted by attaching a bob of mass m to a sting of length L fixed at its uppe end. The bob oscillates in a vetical cicle. It is found that the speed of the bob is v when the sting makes an angle θ with the vetical. Find the tension in the sting at this instant. Sol. The foces acting on the bob ae (figue) (a) the tension T (b) the weight mg. As the bob moves in a vetical cicle with cente at O, the adial acceleation is v / L towads O. Taking the components along this adius and applying Newton s second law, we get T mgcos θ mv / L o, T m(gcos + v / L). Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 0 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com Ex. 4 Pove that a moto ca moving ove a convex bidge is lighte than the same ca esting on the same bidge. Sol. The motion of the moto ca ove a convex bidge AB is the motion along the segment of a cicle AB (Figue; The centipetal foce is povided by the diffeence of weight mg of the ca and the nomal eaction R of the bidge. mv mv mg R o R mg Clealy R < mg, i.e., the weight of the moving ca is less than the weight of the stationay ca. Ex. 5 A body weighing 0.4 kg is whiled in a vetical cicle with a sting making evolutions pe second. If the adius of the cicle is. m. Find the tension (a) at the top of the cicle, (b) at the bottom of the cicle. Given : g 0 m s and π 3.4. Sol. Mass, m 0.4 kg; time peiod second, adius,. m π Angula velocity, ω 4π ad s.56 ad s. / (a) At the top of the cicle,. mv T mg mω mg m(ω g) 0.4 (..56.56 9.8) N 7. N (b) At the lowest point, T m(ω + g) 80 N Ex. 6 You may have seen in a cicus a motocyclist diving in vetical loops inside a death wall (a hollow spheical chambe with holes, so that the cyclist does not dop down when he is at the uppemost point, with no suppot fom below. What is the minimum speed equied at the uppemost position to pefom a vetical loop if the adius of the chambe is 5 m? Sol. When the motocyclist is at the highest point of the death-well, the nomal eaction R on the motocyclist by the ceiling of the chambe acts downwads. His weight mg also act downwads. These two foces ae balanced by the outwad centifugal foce acting on him mv R + mg Hee v is the speed of the motocyclist and m is the mass of the motocyclist (including the mass of the moto cycle). Because of the balancing of the foces, the motocyclist does not fall down. The minimum speed equied to pefom a vetical loop is given by equation () when R 0. mg mv min o v min g o v min g 9.8 5 m s 5.65 ms. So, the minimum speed, at the top, equied to pefom a vetical loop is 5.65 m s. Ex. 7 A simple pendulum is constucted by attaching a bob of mass m to a sting of length L fixed at its uppe end. The bob oscillates in a vetical cicle. It is found that the speed of the bob is v when the sting makes an angle α with the vetical. Find the tension in the sting and the magnitude of net foce on the bob at the instant. Sol. (i) The foces acting on the bob ae : ///////////// (a) the tension T (b) the weight mg α As the bob moves in a cicle of adius L with cente at O. A centipetal foce mv of magnitude is equied towads O. This foce will be povided by the L esultant of T and mg cos α. Thus, o T mg cos α mv T m v gcosα + L L T α mg sin α mg cos α mg Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com (ii) F net (mg sinα) mv + L v m g sin α + L Q. 3 One end of a sting of length.4 m is tied to a stone of mass 0.4 kg and the othe end to a small pivot. Find the minimum velocity of stone equied at its lowest point so that the sting does not slacken at any point in its motion along the vetical cicle? Ans. 8.5 ms Q. 4 A paticle of mass m slides without fiction fom the top of a hemisphee of adius. At what height will the body lose contact with the suface of the sphee? Ans. At a height of /3 above the cente of the hemisphee. CIRCULAR TURNING ON ROADS When vehicles go though tunings, they tavel along a nealy cicula ac. Thee must be some foce which will poduce the equied centipetal acceleation. If the vehicles tavel in a hoizontal cicula path, this esultant foce is also hoizontal. The necessay centipetal foce is being povided to the vehicles by following thee ways.. By fiction only. By banking of oads only. 3. By fiction and banking of oads both. In eal life the necessay centipetal foce is povided by fiction and banking of oads both. Now let us wite equations of motion in each of the thee cases sepaately and see what ae the constant in each case.. BY FRICTION ONLY Suppose a ca of mass m is moving at a speed v in a hoizontal cicula ac of adius. In this case, the necessay centipetal foce to the ca will be povided by foce of fiction f acting towads cente mv Thus, f Futhe, limiting value of f is µn o f L µ N µ mg (N mg) mv Theefoe, fo a safe tun without sliding f L mv o µ mg o v µ o v g g µ Hee, two situations may aise. If µ and ae known to us, the speed of the vehicle should not exceed v µ g and if v and ae known to us, the coefficient of fiction should be geate than. g Q. 5 A bend in a level oad has a adius of 00 m. Calculate the maximum speed which a ca tuning this bend may have without skidding. Given : µ 0.8. Ans. 8 ms. BY BANKING OF ROADS ONLY Fiction is not always eliable at cicula tuns if high speeds and shap tuns ae involved. to avoid dependence on fiction, the oads ae banked at the tun so that the oute pat of the oad is some what lifted compaed to the inne pat. Applying Newton s second law along the adius and the fist law in the vetical diection. mv N sinθ o N cosθ mg fom these two equations, we get v tanθ o g v g tanθ Ex. 8 A cicula tack of adius 600 m is to be designed fo cas at an aveage speed of 80 km/h. What should be the angle of banking of the ack? Sol. Let the angle of banking be θ. The foces on the ca ae (figue) 4 Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com (a) weight of the ca Mg downwad and (b) nomal foce N. Fo pope banking, static fictional foce is not needed. Fo vetical diection the acceleation is zeo. So, Ncos θ Mg...(i) F o h o i z o n t a l d i e c t i o n, t h e a c c e l e a t i o n i s v / towads the cente, so that Nsin θ Mv /...(ii) Fom (i) and (ii), tan θ v / g Putting the values, tan θ θ.6º. 80( km / h) ( 600m)( 0m / s ) 0.467 3. BY FRICTION AND BANKING OF ROAD BOTH If a vehicle is moving on a cicula oad which is ough and banked also, then thee foces may act on the vehicle, of these the fist foce, i.e., weight (mg) is fixed both in magnitude and diection. NOTE : (i) The diection of second foce, i.e., nomal eaction N is also fixed (pependicula o oad) while the diection of the thid foce i.e., fiction f can be eithe inwads o outwads while its magnitude can be vaied upto a maximum limit (f L µn). So the magnitude of nomal eaction N and diections plus magnitude of fiction f ae mv so adjusted that the esultant of the thee foces mentioned above is towads the cente. Of these m and ae also constant. Theefoe, magnitude of N and diections plus magnitude of fiction mainly depends on the speed of the vehicle v. Thus, situation vaies fom poblem to poblem. Even though we can see that : (i) Fiction f will be outwads if the vehicle is at est v 0. Because in that case the component weight mg sinθ is balanced by f. (ii) Fiction f will be inwads if v > g tanθ (iii) Fiction f will be outwads if v < g tanθ and (iv) Fiction f will be zeo if (ii) v g tanθ v The expession tan θ also gives the angle of banking fo an aicaft, i.e., the angle though g which it should tilt while negotiating a cuve, to avoid deviation fom the cicula path. v The expession tan θ also gives the angle at which a cyclist should lean inwad, when g ounding a cone. In this case, θ is the angle which the cyclist must make with the vetical. Ex. 9 A hemispheical bowl of adius R is otating about its axis of symmety which is kept vetical. A small ball kept in the bowl otates with the bowl without slipping on its suface. If the suface of the bowl is smooth and the angle made by the adius though the ball with the vetical is α. Find the angula speed at which the bowl is otating. Sol. Let ω be the angula speed of otation of the bowl. Two foce ae acting on the ball.. nomal eaction N. weight mg The ball is otating in a cicle of adius ( R sin α) with cente at A at an angula speed ω. Thus, N sin α mω mrw sin α...(i) and N cos α mg...(ii) Dividing Eqs. (i) by (ii), we get ω R cosα g ω g Rcosα CENTRIFUGAL FORCE When a body is otating in a cicula path and the centipetal foce vanishes, the body would leave the cicula path. To an obseve A who is not shaing the motion along the cicula path, the body appeas to fly Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 3 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.
FREE Download Study Package fom website: www.tekoclasses.com & www.mathsbysuhag.com Get Solution of These Packages & Lean by Video Tutoials on www.mathsbysuhag.com off tangentially at the point of elease. To anothe obseve B, who is shaing the motion along the cicula path (i.e., the obseve B is also otating with the body which is eleased, it appeas to B, as if it has been thown off along the adius away fom the cente by some foce. This inetial foce is called centifugal foce.) mv Its magnitude is equal to that of the centipetal foce.. Centifugal foce is a fictitious foce which has to be applied as a concept only in a otating fame of efeence to apply N.L in that fame) FBD of ball w..t. non inetial fame otating with the ball. Suppose we ae woking fom a fame of efeence that is otating at a constant, angula velocity ω with espect to an inetial fame. If we analyse the dynamics of a paticle of mass m kept at a distance fom the axis of otation, we have to assume that a foce mω eact adially outwad on the paticle. Only then we can apply Newton s laws of motion in the otating fame. This adially outwad pseudo foce is called the centifugal foce. EFFECT OF EARTHS ROTATION TION ON APPARENT ARENT WEIGHT The eath otates about its axis at an angula speed of one evolution pe 4 hous. The line joining the noth and the south poles is the axis of otation. Evey point on the eath moves in a cicle. A point at equato moves in a cicle of adius equal to the adius of the eath and the cente of the cicle is same as the cente of the eath. Fo any othe point on the eath, the cicle of otation is smalle than this. Conside a place P on the eath (figue). Dop a pependicula PC fom P to the axis SN. The place P otates in a cicle with the cente at C. The adius of this cicle is CP. The angle between the line OM and the adius OP though P is called the latitude of the place P. We have CP OP cosθ o, R cosθ whee R is the adius of the eath. If we wok fom the fame of efeence of the eath, we shall have to assume the existence of pseudo foce. In paticula, a centifugal foce mw has to be assumed on any paticle of mass m placed at P. If we conside a block of mass m at point P then this block is at est with espect to eath. If esolve the foces along and pependicula the cente of eath then N + mω cos θ mg N mg mω cos θ N mg mrω cos θ Ex. 0 A body weighs 98N on a sping balance at the noth pole. What will be its weight ecoded on the same scale if it is shifted to the equato? Use g GM/R 9.8 m/s and the adius of the eath R6400 km. Sol. At poles, the appaent weight is same as the tue weight. Thus, 98N mg m(9.8 m/s ) At the equato, the appaent weight is mg mg m ϖ R π ad The adius of the eath is 6400 km and the angula speed is ϖ 7.7 0 4 60 60s 6 ad/s mg 98N (0 kg) (7.7 0 5 s ) (6400 km) 97.66N Teko Classes, Maths : Suhag R. Kaiya (S. R. K. Si), Bhopal Phone : 0 903 903 7779, 0 98930 5888. page 4 Successful People Replace the wods like; "wish", "ty" & "should" with "I Will". Ineffective People don't.