Non-uniform circular motion *

Transcription

1 OpenSax-CNX module: m Non-uniform circular moion * Sunil Kumar Singh This work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License 2.0 Wha do we mean by non-uniform circular moion? The answer lies in he deniion of uniform circular moion, which denes i be a circular moion wih consan speed. I follows hen ha non-uniform circular moion denoes a change in he speed of he paricle moving along he circular pah as shown in he gure. Noe specially he change in he velociy vecor sizes, denoing change in he magniude of velociy. Circular moion Figure 1: The speed of he paricle changes wih ime in non-uniform circular moion. A change in speed means ha unequal lengh of arc (s) is covered in equal ime inervals. I furher means ha he change in he velociy (v) of he paricle is no limied o change in direcion as in he case * Version 1.7: Nov 19, :39 am hp://creaivecommons.org/licenses/by/2.0/

2 OpenSax-CNX module: m of uniform circular moion. In oher words, he magniude of he velociy ( v) changes wih ime, in addiion o coninuous change in direcion, which is inheren o he circular moion owing o he requiremen of he paricle o follow a non-linear circular pah. 1 Radial (cenripeal) acceleraion We have seen ha change in direcion is accouned by radial acceleraion (cenripeal acceleraion), which is given by following relaion, a R = v2 r The change in speed have implicaions on radial (cenripeal) acceleraion. There are wo possibiliies : 1: The radius of circle is consan (like in he moion along a circular rail or moor rack) A change in v shall change he magniude of radial acceleraion. This means ha he cenripeal acceleraion is no consan as in he case of uniform circular moion. Greaer he speed, greaer is he radial acceleraion. I can be easily visualized ha a paricle moving a higher speed will need a greaer radial force o change direcion and vice-versa, when radius of circular pah is consan. 2: The radial (cenripeal) force is consan (like a saellie roaing abou he earh under he inuence of consan force of graviy) The circular moion adjuss is radius in response o change in speed. This means ha he radius of he circular pah is variable as agains ha in he case of uniform circular moion. In any evenualiy, he equaion of cenripeal acceleraion in erms of speed and radius mus be saised. The imporan hing o noe here is ha hough change in speed of he paricle aecs radial acceleraion, bu he change in speed is no aeced by radial or cenripeal force. We need a angenial force o aec he change in he magniude of a angenial velociy. The corresponding acceleraion is called angenial acceleraion. 2 Angular velociy The angular velociy in non-uniform circular moion is no consan as ω = v/r and v is varying. We consruc a daa se here o have an undersanding of wha is acually happening o angular speed wih he passage of ime. Le us consider a non-uniform circular moion of a paricle in a cenrifuge, whose linear speed, saring wih zero, is incremened by 1 m/s a he end of every second. Le he radius of he circle be 10 m v ω (s) (m/s) (rad/s) The above daa se describes jus a simplied siuaion for he purpose of highlighing variaion in angular speed. We can visualize his change in erms of angular velociy vecor wih increasing magniudes as shown in he gure here :

3 OpenSax-CNX module: m Angular velociy Figure 2: Angular velociy changes wih ime in non-uniform circular moion. Noe ha magniude of angular velociy i.e. speed changes, bu no is direcion in he illusraed case. However, i has been poined ou earlier ha here are acually wo direcional possibiliies i.e. clockwise and ani-clockwise roaion. Thus, a circular moion may also involve change of direcion besides a change in is magniude. 3 Tangenial acceleraion The non-uniform circular moion involves a change in speed. This change is accouned by he angenial acceleraion, which resuls due o a angenial force and which acs along he direcion of velociy circumferenially as shown in he gure. I is easy o realize ha angenial velociy and acceleraion are angenial o he pah of moion and keeps changing heir direcion as moion progresses.

4 OpenSax-CNX module: m Tangenial acceleraion Figure 3: Velociy, angenial acceleraion and angenial force all ac along he same direcion. We noe ha velociy, angenial acceleraion and angenial force all ac along he same direcion. I mus, however, be recognized ha force (and hence acceleraion) may also ac in he opposie direcion o he velociy. In ha case, he speed of he paricle will decrease wih ime. The magniude of angenial acceleraion is equal o he ime rae of change in he speed of he paricle. a T = v (3) Example 1 Problem : A paricle, saring from he posiion (5 m,0 m), is moving along a circular pah abou he origin in xy plane. The angular posiion of he paricle is a funcion of ime as given here, θ = Find (i) cenripeal acceleraion and (ii) angenial acceleraion and (iii)direcion of moion a =0. Soluion : From he daa on iniial posiion of he paricle, i is clear ha he radius of he circle is 5 m. (i) For deermining cenripeal acceleraion, we need o know he linear speed or angular speed a a given ime. Here, we diereniae angular posiion funcion o obain angular speed as : ω = θ =

5 OpenSax-CNX module: m Angular speed is varying as i is a funcion of ime. For = 0, Now, he cenripeal acceleraion is : ω = 0.2 rad / s a R = ω 2 r = ( 0.2 ) 2 x 5 = 0.04 x 5 = 0.2 m / s 2 (ii) For deermining angenial acceleraion, we need o have expression of linear speed in ime. v = ωr = ( ) x 5 = We obain angenial acceleraion by diereniaing he above funcion : a T = v = 10 m / s 2 Evidenly, he angenial acceleraion is consan and is independen of ime. (iii) Since, he angular speed is evaluaed o be posiive a = 0, i means ha angular velociy is posiive. This, in urn, means ha he paricle is roaing ani-clockwise a = 0. 4 Angular acceleraion The magniude of angular acceleraion is he raio of angular speed and ime inerval. α = ω (3) If he raio is evaluaed for nie ime inerval, hen he raio is called average angular acceleraion and If he raio is is evaluaed for inniesimally small period ( 0), hen he raio is called insananeous angular acceleraion. Mahemaically, he insananeous angular acceleraion is : α = ω = 2 θ 2 (3) The angular acceleraion is measured in rad / s 2. I is imporan o emphasize here ha his angular acceleraion is associaed wih he change in angular speed (ω) i.e. change in he linear speed of he paricle (v = ωr) - no associaed wih he change in he direcion of he linear velociy (v). In he case of uniform circular moion, ω = consan, hence angular acceleraion is zero. 5 Relaionship beween linear and angular acceleraion We can relae angular acceleraion (α) wih angenial acceleraion ( a T ) in non uniform circular moion as : a T = v = 2 s 2 = 2 2 ( r θ ) = r 2 θ 2 a T = α r We see here ha angular acceleraion and angenial acceleraion are represenaion of he same aspec of moion, which is relaed o he change in angular speed or he equivalen linear speed. I is only he dierence in he manner in which change of he magniude of moion is described. The exisence of angular or angenial acceleraion indicaes he presence of a angenial force on he paricle.

6 OpenSax-CNX module: m Noe : All relaions beween angular quaniies and heir linear counerpars involve muliplicaion of angular quaniy by he radius of circular pah r o yield o corresponding linear equivalens. Le us revisi he relaions so far arrived o appreciae his aspec of relaionship : s = θr v = ωr a T = αr (3) 6 Linear and angular acceleraion relaion in vecor form We can represen he relaion beween angular acceleraion and angenial acceleraion in erms of vecor cross produc : a T = αxr Tangenial and angular acceleraion Figure 4: Angular acceleraion is an axial vecor. The order of quaniies in vecor produc is imporan. A change in he order of cross produc like ( rxα ) represens he produc vecor in opposie direcion. The direcional relaionship beween hee vecor quaniies are shown in he gure. The vecors a T and r are in xz plane i.e. in he plane of moion, whereas angular acceleraion (α) is in y-direcion i.e. perpendicular o he plane of moion. We can know

7 OpenSax-CNX module: m abou angenial acceleraion compleely by analyzing he righ hand side of vecor equaion. The spaial relaionship among he vecors is auomaically conveyed by he vecor relaion. We can evaluae magniude of angenial acceleraion as : a T = αxr a T = a T = αrsinθ where θ is he angle beween wo vecors α and r. In he case of circular moion, θ = 90, Hence, a T = αxr = αr 6.1 Uniform circular moion In he case of he uniform circular moion, he speed (v) of he paricle in uniform circular moion is consan (by deniion). This implies ha angenial acceleraion, a T, is zero. Consequenly, angular acceleraion ( a Tr ) is also zero. a T = 0 α = 0 7 Descripion of circular moion using vecors In he ligh of new quaniies and new relaionships, we can aemp analysis of he general circular moion (including boh uniform and non-uniform), using vecor relaions. We have seen ha : v = ωxr A close scruiny of he quaniies on he righ hand of he expression of velociy indicae wo possible changes : change in angular velociy ( ω ) and change in posiion vecor (r) The angular velociy ( ω ) can change eiher in is direcion (clockwise or ani-clockwise) or can change in is magniude. There is no change in he direcion of axis of roaion, however, which is xed. As far as posiion vecor ( r) is concerned, here is no change in is magniude i.e. r or r is consan, bu is direcion keeps changing wih ime. So here is only change of direcion involved wih vecor r. Now diereniaing he vecor equaion, we have v = ω Xr + ωx r We mus undersand he meaning of each of he acceleraion dened by he dierenials in he above equaion : The erm " v " represens oal acceleraion ( a ) i.e. he resulan of radial ( a R ) and angenial acceleraion( a T ). The erm ω represens angular acceleraion (α) The erm r represens velociy of he paricle (v)

8 OpenSax-CNX module: m a = αxr + ωxv a = a T + a R where, a T = αxr is angenial acceleraion and is measure of he ime rae change of he magniude of he velociy of he paricle in he angenial direcion and a R = ωxv is he radial acceleraion also known as cenripeal acceleraion, which is measure of ime rae change of he velociy of he paricle in radial direcion. Various vecor quaniies involved in he equaion are shown graphically wih respec o he plane of moion (xz plane) : Vecor quaniies Figure 5 The magniude of oal acceleraion in general circular moion is given by :

9 OpenSax-CNX module: m Resulan acceleraion Figure 6 a = a = ( a T 2 + a R2 ) Example 2 Problem : A a paricular insan, a paricle is moving wih a speed of 10 m/s on a circular pah of radius 100 m. Is speed is increasing a he rae of 1 m / s 2. Wha is he acceleraion of he paricle? Soluion : The acceleraion of he paricle is he vecor sum of muually perpendicular radial and angenial acceleraions. The magniude of angenial acceleraion is given here o be 1 m / s 2. Now, he radial acceleraion a he paricular insan is : a R = v2 r = = 1 m / s2 Hence, he magniude of he acceleraion of he paricle is : a = a = ( a T 2 + a R2 ) = ( ) = 2 m / s 2

10 OpenSax-CNX module: m Exercises Exercise 1 (Soluion on p. 12.) A paricle is moving along a circle in yz - plane wih varying linear speed. Then (a) acceleraion of he paricle is in x direcion (b) acceleraion of he paricle lies in xy plane (c) acceleraion of he paricle lies in xz plane (d) acceleraion of he paricle lies in yz plane Exercise 2 (Soluion on p. 12.) A paricle is moving along a circle of radius r. The linear and angular velociies a an insan during he moion are v and ω respecively. Then, he produc vω represens : (a) cenripeal acceleraion (b) angenial acceleraion (c) angular acceleraion divided by radius (d) None of he above Exercise 3 (Soluion on p. 12.) Which of he following expression represens he magniude of cenripeal acceleraion : (a) 2 r v θ (b) 2 (c) r (d) None of hese Exercise 4 (Soluion on p. 12.) A paricle is circling abou origin in xy-plane wih an angular speed of 0.2 rad/s. Wha is he linear speed (in m/s) of he paricle a a poin specied by he coordinae (3m,4m)? (a)1 (b) 2 (c) 3 (d) 4 Exercise 5 (Soluion on p. 13.) A paricle is execuing circular moion. Man The velociy of he paricle changes from zero o (0.3i + 0.4j) m/s in a period of 1 second. The magniude of average angenial acceleraion is : (a) 0.1 m / s 2 (b) 0.2 m / s 2 (c) 0.3 m / s 2 (d) 0.5 m / s 2 Exercise 6 (Soluion on p. 13.) The radial and angenial acceleraions of a paricle in moions are a T and a R respecively. The moion can be circular if : (a) a R 0, a T 0 (b) a R = 0, a T = 0 (c) a R = 0, a T = (d) a R =, a T = 0 Exercise 7 (Soluion on p. 13.) Which of he following pair of vecor quaniies is/are parallel o each oher in direcion? (a) angular velociy and linear velociy (b) angular acceleraion and angenial acceleraion (c) cenripeal acceleraion and angenial acceleraion (d) angular velociy and angular acceleraion Exercise 8 (Soluion on p. 13.) A paricle is moving along a circle in a plane wih axis of roaion passing hrough he origin of circle. Which of he following pairs of vecor quaniies are perpendicular o each oher : (a) angenial acceleraion and angular velociy (b) cenripeal acceleraion and angular velociy (c) posiion vecor and angular velociy (d) angular velociy and linear velociy

11 OpenSax-CNX module: m Exercise 9 (Soluion on p. 13.) A paricle is execuing circular moion along a circle of diameer 2 m, wih a angenial speed given by v = 2. (a) Tangenial acceleraion direcly varies wih ime. (b) Tangenial acceleraion inversely varies wih ime. (c) Cenripeal acceleraion direcly varies wih ime. (d) Cenripeal acceleraion direcly varies wih square of ime.

12 OpenSax-CNX module: m Soluions o Exercises in his Module Soluion o Exercise (p. 10) The gure here shows he acceleraion of he paricle as he resulan of radial and angenial acceleraions. The resulan acceleraion lies in he plane of moion i.e yz plane. Circular moion Figure 7 Hence, opion (d) is correc. Soluion o Exercise (p. 10) The given produc expands as : vω = v x v r This is he expression of cenripeal acceleraion. Hence, opion (a) is correc. Soluion o Exercise (p. 10) v θ The expression represens he magniude of oal or resulan acceleraion. The dierenial represens he magniude of angular velociy. The expression r represens he magniude of angenial velociy. The expression 2 r is second order diereniaion of posiion vecor (r). This is acually he 2 expression of acceleraion of a paricle under moion. Hence, he expression 2 r represens he 2 magniude of oal or resulan acceleraion. Hence, opion (d) is correc. = v2 r θ

13 OpenSax-CNX module: m Soluion o Exercise (p. 10) The linear speed v is given by : v = ωr Now radius of he circle is obained from he posiion daa. Here, x = 3 m and y = 4 m. Hence, r = ( ) = 5 m v = 0.2 x 5 = 1 m / s Hence, opion (a) is correc. Soluion o Exercise (p. 10) The magniude of average angenial acceleraion is he raio of he change in speed and ime : Now, a T = v v = a T = 0.5 m / s 2 ( ) = 0.25 = 0.5 m / s Hence, opion (d) is correc. Soluion o Exercise (p. 10) Cenripeal acceleraion is a requiremen for circular moion and as such i should be non-zero. On he oher hand, angenial acceleraion is zero for uniform circular acceleraion and non-zero for non-uniform circular moion. Clearly, he moion can be circular moion if cenripeal acceleraion is non-zero. Hence, opions (a) and (d) are correc. Soluion o Exercise (p. 10) The opion (d) is correc. Soluion o Exercise (p. 10) Clearly, vecor aribues in each given pairs are perpendicular o each oher. Hence, opions (a), (b), (c) and (d) are correc. Soluion o Exercise (p. 10) Tangenial acceleraion is found ou by diereniaing he expression of speed : a T = v ( 2 ) = 2 m / s The angenial acceleraion is a consan. Now, le us deermine cenripeal acceleraion, The opion (d) is correc. a R = v2 r = 42 1 = 4 2