ROTATORY MOTION HORIZONTAL AND VERTICAL CIRCULAR MOTION

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Josephine Lawson
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1 ROTATORY MOTION HORIZONTAL AND VERTICAL CIRCULAR MOTION POINTS TO REMEMBER 1. Tanslatoy motion: Evey point in the body follows the path of its peceding one with same velocity including the cente of mass.. Rotatoy motion: Evey point move with diffeent velocity with espect to the axis of otation. The paticle on the axis of otation will have zeo velocity. 3. Cicula Motion: If the motion of a paticle about a fixed point is a cicle, then it is called cicula motion. 4. Axis of otation: Duing otational motion, all the paticles of a igid body move in diffeent cicles and the cente of all these cicles lie on a line called axis of otation. 5. Radius vecto: - Any instant of time, the line joining the position of otating paticle with the cente of otation is called adius vecto at that instant. 6. Radian definition: - The angle subtended at the cente of a cicle by an ac of length equal to its adius is called adian. π adians = Radian = Angula displacement ( θ ): The angle though which the adius vecto of a otating paticle otates in a given time inteval is called angula displacement. 0 Unit: adian Dimension: M L T 8. Angula velocity ( ω): The ate of change of angula displacement of a otating paticle is called angula velocity. dθ Angula velocity( ω ) = ) Unit: ad sec -1 Dimensions: M L T 9. Angula acceleation ( α ) : The ate of change of angula velocity of a otating paticle is called angula acceleation. dω Angula acceleation ( α ) = Unit : ad sec - Dimension: M L T 10. Centipetal foce: A eal foce which is acquied by a body to pefom cicula motion and which acts towads the cente of the cicle is called a centipetal foce. m Cente petal foce = = mω 11. Centifugal foce: A pseudo foce which acts adially outwads fom the cente of the cicula path is called centifugal foce. It is equal in magnitude to the centipetal foce. mv Centifugal foce= = mω

2 1. If a body of mass m is otated in a vetical cicle of adius tension in the sting at mv any point of the vetical cicle is given by T = + mg cosθ Velocity at any point v = g ( 3+ cosθ ) 13. Tension will be maximum at the lowest point Tension will be minimum at the highest point T max T mv = + mg min m v = mg 14. Fo a body of mass m is otated in a vetical cicle of adius, the velocity at the lowest point v = 5g and at the highest point the velocity is v = g Tanslatoy motion v = u + at 1 s = ut + at v = u + as u + v s = a sn = u + ( n ) Rotatoy motion ω = ω + α 1 t 1 θ = ω1t + αt ω = ω + αθ θ = 1 ω + ω 1 α θn = ω1 + ( n ) Shot Answe Questions: 1. Explain the tems: Cicula motion, angula displacement, angula velocity and angula acceleation. A. Cicula Motion: If the motion of a paticle about a fixed point is a cicle, then it is called cicula motion. Angula displacement ( θ ) : The angle though which the adius vecto of a otating paticle otates in a given time inteval is called angula displacement. Angula velocity ( ω): The ate of change of angula displacement of a otating paticle is called angula velocity. dθ Angula velocity ( ω ) = Angula acceleation ( α ): The ate of change of angula velocity of a otating paticle is called angula acceleation. dω Angula acceleation ( α ) =. a) Define angula velocity (b) State its units and dimensions c) State whethe it is a scala o a vecto. If it is a vecto given its diection. A. a) The ate of change of angula displacement of a otating body is called angula dθ velocity angula velocity ( ω ) = b) Unit: ad sec -1 Dimensions: M L T

3 c) Angula velocity is a vecto quantity whose diection is pependicula to the plane of otation and it can be known fom ight hand thumb ule. 3. a) Define angula acceleation. b) State its units and dimension. c) State withe it is a vecto (o) scala. If it is a vecto give its diection. A. a) The ate of change of angula velocity of a otation body is called angula acceleation. dω Angula acceleation ( α ) = b) Unit : ad sec - Dimension: M L T c) Angula acceleation is a vecto quantity whose diection is along the diection of change in the angula velocity. If the angula velocity inceases, the diection is along the diection of angula velocity. If the angula velocity deceases, the diection is opposite to that of angula velocity. 4. Deive v = ω A. Conside a paticle moving in a cicula path of adius with a constant angula velocity ω. Let the paticle moves fom A to B in a small time inteval δt and δθ be a small angula displacement. The instantaneous angula velocity ' ω' at any instant of time t is given by δθ dθ ω = Lim = δt 0 δ t Now, the linea velocity of the paticle v, is given by AB δθ v = Lim = Lim δt 0 δt δt 0 δt δθ dθ = Lim = δt 0 δt v = ω Vectoially v = ω 5. What is unifom cicula motion? Obtain an expession fo the acceleation of a paticle pefoming unifom cicula motion. A. A paticle is said to be in unifom cicula motion if its angula velocity is unifom (o) linea speed is constant thoughout its motion along the cicle. Ex: motion of the moon aound the eath. Centipetal acceleation: Conside a paticle moving in a cicula path of adius with cente at O in the anti clock wise diection. At any instant of time t the position vecto otates though an angle θ fom its initial position along X-axis so that θ = ω t. = cos ω t i + sin ωt j ( ) ( ) (o) = ( cos ω t) i + ( sin ωt) j d But v = = ω ( sin ω t) i + ( cos ωt) j Now acceleation a is given by dv a = = ω ( cos ωt) i ( sin ωt) j

4 = ω ω + ω ( ) ( ) cos t i sin t j cos t i sin t j which is the unit The diection of acceleation is given by ( ω ) + ( ω ) vecto opposite to. Hence the centipetal (o) nomal acceleation is towads the cente of the cicula path. 6. Define centipetal foce. Explain its impotance with suitable examples. A. A eal foce which is acquied by a body to pefom cicula motion and which acts towads the cente of the cicle is called a centipetal foce. m Cente petal foce = = mω Impotance: Centipetal foce is a net (o) unbalanced foce towads the cente of the cicle with out which the paticle moves along the tangent to the cicle. Examples: 1) The gavitational foce between the moon and the eath supplies the necessay centipetal foce fo the moon to pefom a unifom cicula motion. ) While taking a tun on the oad, the fiction between the oad and tyes of a ca povides the necessay centipetal foce. 3) When a stone tied to a thead is otated in a cicula path, the tension in the thead supplies the necessay centipetal foce. 4) While taking a tun in ai fo an aeo plane, the lift povided by its banked wings supplies the necessay centipetal foce. 7. Define centifugal foce. Explain its impotance with suitable examples. A. A pseudo foce which acts adially outwads fom the cente of the cicula path is called centifugal foce. It is equal in magnitude to the centipetal foce. mv Centifugal foce = = mω Impotance : Centifugal foce is a pseudo (o) fictious foce felt. when we ae in a non inetial fame. Examples: 1) Due the spin motion of the eath, the centifugal foce is maximum at the equato and minimum at the poles. Hence the weight of the body is maximum at the poles and minimum at the equato. ) Centifuge is a device in which centifugal foce sepaates lighte and heavy paticles. A centifuge is used to sepaate ceam fom milk (o) to sepaate substance of diffeent masses. 3) A centifugal goveno contols the speed of an engine by using spinning masses that espond to centifugal foce poduced by the engine. 8. Deduce the elation fo the tension on the sting of an object in vetical cicula motion when it is at the lowest point and (ii) highest point of the cicle. A. Let a body of mass m is tied to a sting and whiled in a vetical cicle of adius. At the lowest point (A) Let T 1 be the tension and v 1 be the velocity when the body is at the lowest point of the vetical cicle. The foces on the body ae i) Tension T 1 in the upwad diection

5 ii) Weight mg of the body acting downwads. The esultant of these supplies the necessay centipetal foce mv1 mv1 T1 mg = (O) T1 = + mg At the highest point (B) Let T be the tension and v be the velocity of the body when it is at the highest point of the vetical cicle. The foces on the body ae : i) Tension T 1 acting down wads ii) Weight of the body mg downwads. mv mv T + mg = O T = mg 9. Show that the diffeence in the tensions is 6mg whee mg is the weight of a block pefoming vetical cicula motion unde gavity. A. Let a body of mass m is tied to a sting and whiled in a vetical cicle of adius. When the body is at the highest point of the vetical cicle mv Tension T = mg and Velocity v = g At the lowest point of the vetical cicle mv1 Tension T1 = + mg and Velocity v1 = 5g Now the diffeence in tensions is given by m T 1 T = ( v1 v ) mg + = m ( 5g g ) + mg T1 T = 6mg Vey Shot Answe Questions 1. What is axis of otation? A. Duing otational motion, all the paticles of a igid body move in diffeent cicles and the cente of all these cicles lie on a line called axis of otation. The axis of otation may lie in o outside of the body. Fo the eath moving aound itself the axis passes though the cente of the eath whee as fo the motion of the eath aound the sun the axis passes though the cente of the sun.. Which of the following ae vectos? i) Angula displacement, ii) Angula Velocity, iii) Angula acceleation IV) Moment of inetia. A. Angula displacement, angula velocity and angula acceleation ae pseudo vectos Moment of inetia is a scala. 3. Give the vecto fom of the elation between v and ω A. V = ω

6 4. A igid body is moving in a vetical plane. If its velocity about the lowest point of the Cicle is 5g. What is its velocity when it eaches hoizontal position? A. (PE + KE) at the lowest point = (PE + KE) at the hoizontal point 1 ( ) 1 0+ m 5g = mv + mg v = 3g Execise 1 1. Find the aveage angula velocity of the second s hand of a watch if the second s hand of a watch completes one evolution in 1 minute.sol : n = 1, t = 1 minute = 60 s. π n π 1 π Angula velocity ( ω) = = = ad s t Find the aveage angula velocity of the spinning motion of the eath. Sol: Fo one complete otation of the eath t = 4 hou =86400 seconds π n π 1 π Angula velocity ( ω) = = = ad s. t 86, , When a moto cyclist takes a U-tun in 4 seconds what is the aveage angula velocity of the moto cyclist? Sol: Angula displacement ( θ ) = π ad and t = 4 s. θ π Aveage angula velocity ( ω) = = = ad s. t When the angula displacement of a paticle is given by θ = t + t + t +1find its (i) angula velocity and (ii) angula acceleation at t = second. 3 Sol: The angula displacement, θ = t + t + t +1 and t = s dθ Angula velocity, ω = = 3t + t + 1 = 17 ad s dω Angula acceleation, α = = 6t +. = 14 ad s. 5. A ca is moving in a cicula path with a unifom speed v. Find the magnitude of change in its velocity when the ca otates though an angle θ. Sol: The magnitude of change in velocity = θ = vsin. v + u uv cosθ ( u = v) 6. What is the linea velocity of a peson at the equato of the eath due to its spinning motion? (Radius of the eath = 6400 km.) π n π 1 π Sol : Angula velocity of the eath, ω = = = ad s. t , 00 6 R = 6400 km = m 6 π The linea velocity, v = Rω = = m / s 43, 00

7 7. A ball of 00 g is on one end of a sting of length 0 cm. It is evolved in a hoizontal cicle at an angula fequency of 6 pm. Find (i) the angula velocity, (ii) the linea velocity, (iii) the centipetal acceleation, (iv) the centipetal foce and (v) the tension on the sting. Sol: m = 0. kg ; = 0. m; n = 6; t = 1 minute = 60 s. π n π 6 π (i) Angula velocity, ω = = = = ad s. t 60 5 (ii) Linea velocity, v = ω = = 0.157m s. (iii) Centipetal acceleation, ω ( ) a = = = m s. (iv) Centipetal foce, = = ( ) c F mω = N. c (v) The tension on the sting = centipetal foce = N. 8. Consideing figue, (a) with what angula speed ' ω' must m with a adius otate on a fictionless table so that M does not move? (b) If m = 1.0 kg, M = kg and = 0.5 m, find ω. Sol : (a) The tension on the sting (Mg) supplies the necessay centipetal foce to the body of mass m evolving in a cicula obit of adius. T = mω ( o) m ω = M g Angula speed = ω = Mg. m b) m = 1.0 k.g ; M = 10.0 kg ; = 0.5 m ; Mg ω = = = = m g = ad s. 9.8 m s. 9. A stone tied to the end of a sting is whiled in a hoizontal cicle. The mass of the stone is 1.0 kg and the sting is 0.50 m long. If the stone evolves at a constant speed of 10 times in s, (a) what is the tension on the sting? (b) What would happen to the tension on the sting if the mass was doubled and all the othe quantities stayed the same? (c) What would happen to the tension on the sting if the peiod was doubled and all the othe quantities emain the same? Sol : m =1 kg; = 0.5 m; n = 10; t = s. π n π The angula velocity, ω = = = = 4 ad s. t a) The tension on the sting = mω ( ) = = 8 N. b) Keeping and ω. constant, if m is doubled, the tension on the sting becomes double. i.e. T= 16 N. c) Keeping m and constant, if time peiod is doubled, π Befoe doubling, the tension = Tension = mω = m = 8N T π 8 Afte doubling, the tension = Tension = mω = m = = N T 4 Hee T is the time peiod.

8 10. A stone of mass.0 kg is tied to the end of a sting of m length. It is whiled in a hoizontal cicle. If the beaking tension of the sting is 400N, calculate the maximum velocity of the stone. Sol : m =.0 kg; = m; tension = 400 N. At the beaking tension, let v be the maximum velocity of the stone. 400 = mv v 400 = v = 0 m s. 11. A mass m is evolving in a vetical cicle at the end of a sting of length. Calculate the diffeence in kinetic enegies at the top and bottom of the cicle. Sol: If the velocities of the body at the top and bottom ae and espectively. But, v = u + as vb = vt g u = vl, v = vh, a = g, s = Since the body goes fom the bottom to the top of the cicle. vt vb = 4g 1 1 Change in kinetic enegy = = m( vt vb ) = m 4g = mg 1. A pendulum bob of mass m is held out in the hoizontal position and then eleased fom est. If the sting is of length l, what is the velocity of the bob and tension on the sting when the bob eaches the lowest position? Sol : Conside the lowest position as efeence position i) Fom the law of consevation of enegy, Total enegy of the bob in the hoizontal = Total enegy of the bob at the lowest position. (KE + PE) in the hoizontal position = (KE + PE) at the lowest position mgl = mv + 0 v = gl. ii) The tension in the sting at the lowest position mv m( gl) = + mg = + mg = 3mg. l 13. A body is allowed to slide down a fictionless tack fom est position at its top unde gavity. The tack ends in a cicula loop of diamete D. What is the minimum height of the inclined tack in tems of D so that it may complete successfully the loop? Sol : Conside the end of the tack is taken as efeence position Fom the law of consevation of enegy, Total enegy of the body at the top of the inclined tack = Total enegy of the body at the end of the tack. (KE + PE) at the top = (KE + PE) at the bottom mgh = mv + 0 v = gh Whee h is the height of the inclined tack. 5 5D D 5g = gh h = = = 4 To have a complete the loop,

9 14. A 300 g ball on a 60 cm long sting is swung in a vetical cicle whose lowest point is 00 cm above the floo. The tension in the sting when the ball is at the vey bottom of the cicle is 5.10 N. A vey shap knife is suddenly inseted to cut the sting diectly below the point of suppot. How fa does the ball tavel befoe if hits the floo? Sol : M = 0.3 kg; = 0.6 m ; h = m ; tension = 5.1 N; g = 9.8 ms. Tension at the bottom = v l =.078 m s. Mv l v + Mg 5.10 = l This velocity is in the hoizontal diection and hence the ball becomes a hoizontal pojectile. Range of the hoizontal pojectile h R = vlt = vl =.078 g 9.8 = 1.38m Execise 1 1. A ca coves a semi cicula tun in 10 seconds. What is the aveage angula velocity of the ca? A.. Angula displacement = π adian Time taken = 10s angula displacement π Angula velocityω = = = 0.341ad s time The angula displacement of a paticle is θ = 3t + 9t + 6t + 4 whee θ is in adian and t is in second. Find its i) angula velocity and ii) angula acceleation at t = 1 s. 3 A. Angula displacement θ = 3t + 9t + 6t + 4 (i) Angula velocity ω = d θ = 9t + 18t + 6 At t = 1, ω = = 33 ad s 1 Angula acceleation At t = 1, a = 18 (1) + 18 = 36 ad s 3. What is the angula displacement of the seconds hand of clock in 45 second? A. Angula displacement of seconds hand in 60 s = π ad Angula displacement in 45 s = π = 3 π ad 4. To simulate the acceleation of lage ockets, astonauts ae spun at the end of a long otating beam of adius 9.8 m. What angula velocity is equied to geneate a centipetal acceleation 8 times the acceleation due to gavity? A. Centipetal acceleation = ω = 9.8 m and ω = 8g o ω = =.88 ad s 1

10 5. A 0.5 kg mass is otated in a hoizontal cilce of adius 0 cm. Calculate the centipetal foce acting on it, if its angula speed of evolution is 0.6 ad S A. m = 0.5 kg, = 0 x 10 m, ω = 0.6 ad s 1 Centipetal foce = m = 0.5 x 0 x 0.36 = N A cetain sting beaks unde a tension of 45 kg wt. A mass of 100 g is attached to this sting of length 50cm and is whiled in a hoizontal cicle. Find the maximum numbe of evolutions possible pe second without beaking the sting. A. m = 100 g = 0.1 kg, l = 50 cm = 0.5 m Tension in the sting = = 0.1 x 0.5 xω whee ω is the angula velocity of the mass But, ml ω = 45 g 0.1 x 0.5 x 4π x n = 45g o n 45g = = π o n = 15 ev/s 7. If the centipetal foce acting on a body evolving along a cicula path of adius 5 m is 00 N find its kinetic enegy.(may009) mv A. Centipetal foce = and = 5 m mv = 00 = K.E = mv = = 500 J 8. In a moto cycle stunt called well of death a ide dives ound the inne wall of a hollow cylindical chambe. If the diamete of the cylindical chambe is 5 m, what should be the minimum angula speed of the ide to pevent him fom sliding down? ( µ = 0.8) A.. Radius of the chambe = 5 m mv Nomal eaction on the ide = R = = mω Whee is the minimum angula speed equied to pevent the ide fom sliding down Fictional foce = µ R = µ m g 9.8 µ mω = mg o ω = = = 0.98 = 0.49 o µ ω = 0.7 ad s 9. A moto cyclist ides in vetical cicles in a hollow sphee of adius 5 m. Find the minimum speed equied so that he does not lose contact with sphee at the highest point. A. = 5m, g = 9.8 ms Citical speed at the highest point = g = = 7ms

11 10. A bucket containing wate is tied to end of a ope 0.5 m long and otated about the othe end in a vetical cicle. Find the minimum numbe of otations the bucket can make pe minute so that the wate in the bucket does not spill. A. The weight of wate should povide the necessay centipetal foce. ie., π n gt mω mg m mg n = = = = = 4.4 pm t 4π 11. A coin is kept a distance of 10 cm fom the cente of a cicula tun table. If the coefficient of static fiction between the table and the coin is 0.8 find the fequency of otation of the disc at which the coin will just begin to slip. A. Fo the coin just not to slip, the equied condition is m ω = µ mg o o m 4π n = µ g µ g n = = = 4π 4π 0.1 o n = ev/s = = ev/min 1. A stone of mass 0.5 kg is tied to one end of a sting of length 1. m and whiled in vetical cicula path. Find the maximum and minimum tension in the sting, if the speed of the stone at the lowest point is 8 ms -1 A. m = 0.5 Kg, l = 1. m V 1 = 8 ms 1 V V1 = g s O V 64 = ( 9.8).4 O V = mv Tmax = + mg = = = 31.57N l mv Tmin = mg = = 4.9 =.166N l A stone of mass 1 kg is tied to one end of a sting of length 50 cm. It is whiled in a vetical cicle. If it can withstand a maximum tension of 58.8 N what is the velocity of the stone at the top of the vetical cicle? A. m = 1 Kg, l = 50 cm = 0.5 m mv T max = T 1 = mg + 1 l whee V 1 is the velocity at the bottom most point of the vetical cicle 1 v = 1 x o v1 = ms 0.5 Let V be the velocity at the top most point V V1 = g x l o V 49/ = 4 x 9.8 x ½ V = 4.9 =.14 ms 1

12 14. A stone of mass kg is tied to one end of a sting of length 50 cm. It is whiled in a vetical cicle. If the velocity of the stone at the top of the cicle is 5.0m s -1 calculate the tension in the sting at a) the top of the cicle and b) the bottom of the cicle. A. m = Kg, l = 50 cm = 0.5 m V = 50 ms 1 V V1 = g l o V1 = = 44.6 a) b) mv 5 T = mg = 9.8 = = 80.4N l 0.5 mv T1 = + mg = = = 198.0N l The angula fequency of a fan inceases fom 30 pm to 60 pm in 3.14 s. A dust paticle is pesent on the blades of the fan at a distance of 0 cm fom the axis of otation. Find i) the aveage angula acceleation ii) the adial acceleation of the dust paticle at the end of 3.14 s, iii) the tangential acceleation of the dust paticle at the end of 3.14 s and iv) the net acceleation of dust paticle at the end of 3.14 s. 1 1 A.. ω 0 = 30/60 ps = π = π ads ω = 60/60 ps = π ad s 1 t = 3.14 s = π s Distance of dust paticle fom cente = 0 cm = 0. m i) Let α be the angula acceleation Then ω= ω 0 + α t o π = π + (α t) o α = 1 ad s v ii) Radial acceleation a = = ω whee ω is the angula velocity at the end of 3.14 s = π s This is equal to p ad s 1 a = ω = 0. π = 0.8π = 7.90 ms ( ) iii) Tangential acceleation iv) Radial acceleation a t = α = 0. 1 = 0. ms a = a + a = = 7.903m s t

13 ASSESS YOURSELF 1) A body is in pue otation. The linea speed v of a paticle of the body, the distance of the paticle fom the axis and the angula velocity ω of the body v ae elated as ω = Does this mean that the angula velocity is invesely popotional to the distance of the paticle fom the axis? A) No, In pue otation the angula velocity (ω) of a body is independent of distance of paticle fom the axis of otation i.e. all paticles in a otating body move with same angula velocity. ) What is the magnitude of ω. v? A) Zeo, angula velocity vecto, linea velocity vecto ae pependicula to each othe 3) What is the magnitude of ω.? A) Zeo, because angula velocity vecto and adius vecto ae pependicula to each othe. 4) Accoding to Boh s theoy, an electon evolves ound the nucleus in a cicula obit in an atom with constant angula velocity. Which supplies the necessay centipetal foce? A) Electostatic foce of attaction between the positively chaged nucleus and evolving electon of negative chage acting towads the nucleus suppliesthe necessay centipetal foce. 5) Is the kinetic enegy of a paticle pefoming unifom cicula motion constant? A) Yes, because in unifom cicula motion magnitude of linea velocity (v) emains constant. 6) Which physical quantities ae constant in unifom cicula motion? A) Angula velocity, magnitude of linea velocity, kinetic enegy and angula momentum emains constant. 7) Which physical quantities ae vaiable in unifom cicula motion? A) The diection of linea velocity, linea momentum, centipetal acceleation and centipetal foce vaiable. 8) A body moves along a cicula path with vaiable angula velocity. Does the body have only centipetal acceleation? A) A body moving along a cicula path with vaiable angula velocity possesses both centipetal acceleation a v = along the adius towads the cente and tangential acceleation is given by a = a + a = + ( α ) c a t t c = α along the tangent. Hence the esultant acceleation v

14 9) The dive of a ca taveling with a velocity v suddenly sees a boad wall ahead of him at a distance a. Is it bette fo him to beak o tun shaply in a cicle? A) Beaking equies less fiction than tuning shaply in a cicle.hence baking is pefeed 10) An atificial satellite moving ound the eath in a cicula obit possesses a changing acceleation. Is it tue? A) Yes, The acceleation is constant in magnitude but changes its diection. 11) A bottle of soda wate is gasped by the neck and otated in a vetical cicle. Nea which potion of the bottle do the bubbles collect? A) Nea the neck 1) When a moto cyclist moves in a vetical cicle aound a spheical globe without falling down at the top most point which balances the gavitational foce? A) Centifugal foce.

Physics 4A Chapter 8: Dynamics II Motion in a Plane

Physics 4A Chapter 8: Dynamics II Motion in a Plane Physics 4A Chapte 8: Dynamics II Motion in a Plane Conceptual Questions and Example Poblems fom Chapte 8 Conceptual Question 8.5 The figue below shows two balls of equal mass moving in vetical cicles.