Solution to Theoretical uetion art Swing with a Falling Weight (a Since the length of the tring Hence we have i contant, it rate of change ut be zero 0 ( (b elative to, ove on a circle of radiu with angular velocit, o v tˆ t ˆ ( (c efer to Fig elative to, the diplaceent of in a tie interval t i r ( ( rˆ ( ˆ t [( ( rˆ tˆ ] t It follow v rˆ t ˆ ( r tˆ rˆ Figure (d The velocit of the particle relative to i the u of the two relative velocitie given in Eq ( and ( o that v v v ( rˆ tˆ tˆ rˆ (4 (e efer to Fig The ( tˆ -coponent of the velocit change v i given b t v v v ( ˆ t Therefore, the tˆ -coponent of the acceleration a v / t i given b tˆ aˆ v Since the peed v of the particle i according to Eq (4, we ee that the tˆ -coponent of the particle acceleration at i given b a tˆ v ( (5 rˆ tˆ v v v v v v v Figure
Note that, fro Fig, the radial coponent of the acceleration a alo be obtained a a rˆ dv / dt d( / dt (f efer to Fig The gravitational potential energ of the particle i given bu gh It a be epreed in ter of and a U ( g[ co in ] (6 h in co co Figure (g t the lowet point of it trajector, the particle gravitational potential energ U ut aue it iniu value U B differentiating Eq (6 with repect to and uing Eq (, the angle correponding to the iniu gravitational energ can be obtained du d g in in co d d g[ in ( in co ] gco t du, 0 We have d The lowet point of the particle trajector i hown in Fig 4 where the length of the tring egent of i / Figure 4 Fro Fig 4 or Eq (6, the iniu potential energ i then U / g[ ( / (7 U ( ] Initiall, the total echanical energ E i 0 Since E i conerved, the peed v of the particle at the lowet point of it trajector ut atif
E v 0 U (8 Fro Eq (7 and (8, we obtain v U / g[ ( / ] (9 art B (h Fro Eq (6, the total echanical energ of the particle a be written a E 0 v U ( v g[ co in ] (B Fro Eq (4, the peed v i equal to Therefore, Eq (B iplie v ( g[ co in ] (B et T be the tenion in the tring Then, a Fig B how, the tˆ -coponent of the net force on the particle i T g in Fro Eq (5, the tangential acceleration of the particle i ( Thu, b Newton econd law, we have ( T g in (B T g in g Figure B ccording to the lat two equation, the tenion a be epreed a g T ( g in [ co in ] g [tan ( ](in g ( (in (B4 The function tan( / and ( / / are plotted in Fig B 4
0 Figure B 0 0 0-0 -0 tan ( tan -0 0 / / 5 / Fro Eq (B4 and Fig B, we obtain the reult hown in Table B The angle at which i called ( < < and i given b ( tan (B5 or, equivalentl, b tan (B6 Since the ratio / i known to be given b 9 cot ( tan ( (B7 8 6 8 8 one can readil ee fro the lat two equation that 9 / 8 Table B ( in tenion T 0 < < poitive poitive poitive 0 poitive < < negative negative poitive zero negative zero < < poitive negative negative Table B how that the tenion T ut be poitive (or the tring ut be taut and traight in the angular range 0< < nce reache, the tenion T becoe zero and the part of the tring not in contact with the rod will not be traight afterward The hortet poible value in for the length of the line egent therefore occur at and i given b 5
9 9 in ( cot cot 5 (B8 8 6 8 6 When, we have T 0 and Eq (B and (B then lead to v gin in Hence the peed v i g 4g v gin in cot in co 6 8 6 (B9 g (i When, the particle ove like a projectile under gravit hown in Fig B, it i projected with an initial peed v fro the poition (, in a direction aking an angle φ ( / with the -ai The peed v H of the particle at the highet point of it parabolic trajector i equal to the -coponent of it initial velocit when projected Thu, 4g vh v in( co in 0 44 g (B0 6 8 The horizontal ditance H traveled b the particle fro point to the point of aiu height i v H in ( v 9 in 0455 g g 4 (B φ v H in v H (, Figure B The coordinate of the particle when are given b co in in co in in 0 58 (B 8 8 in in co in in co 478 (B 8 8 Evidentl, we have > ( H Therefore the particle can indeed reach it aiu height without triking the urface of the rod 6
art C (j ue the weight i initiall lower than b h a hown in Fig C h Figure C When the weight ha fallen a ditance D and topped, the law of conervation of total echanical energ a applied to the particle-weight pair a a te lead to gh E g( h D (C where E i the total echanical energ of the particle when the weight ha topped It follow E gd (C et Λ be the total length of the tring Then, it value at 0 ut be the ae a at an other angular diplaceent Thu we ut have Λ h ( ( h D (C Noting that D α and introducing l D, we a write D ( α (C4 Fro the lat two equation, we obtain D (C5 fter the weight ha topped, the total echanical energ of the particle ut be conerved ccording to Eq (C, we now have, intead of Eq (B, the following equation: [ co in ] E gd v g (C6 The quare of the particle peed i accordingl given b gd v ( g ( co in (C7 Since Eq (B till applie, the tenion T of the tring i given b T g in ( (C8 Fro the lat two equation, it follow 7
8 in co ( in co ( in ( D g D g g T (C9 where Eq (C5 ha been ued to obtain the lat equalit We now introduce the function in co ( f (C0 Fro the fact D >> (, we a write in( co in ( φ f (C where we have introduced (, tan φ (C Fro Eq (C, the iniu value of f( i een to be given b in f (C Since the tenion T reain nonnegative a the particle wing around the rod, we have fro Eq (C9 the inequalit 0 ( in f D (C4 or (C5 Fro Eq (C4, Eq (C5 a be written a ( α (C6 Neglecting ter of the order (/ or higher, the lat inequalit lead to α (C7 The critical value for the ratio D/ i therefore c α (C8
Total Score art 4 pt art B 4 pt Sub Score (a 05 (b 05 (c 07 (d 07 (e 07 (f 05 (g 07 (h 4 arking Schee Theoretical uetion Swing with a Falling Weight arking Schee for nwer to the roble elation between and ( 0 for 0 for proportionalit contant (- Velocit of relative to ( tˆ 0 for agnitude 0 for direction tˆ article velocit at relative to ( v rˆ tˆ 00 for agnitude and direction of rˆ -coponent v 00 for agnitude and direction of tˆ -coponent article velocit at relative to ( v v v rˆ 0 for vector addition of v and v 00 for agnitude and direction of v tˆ -coponent of particle acceleration at 0 for relating a or a tˆ to the velocit in a wa that iplie a tˆ v / 04 for a tˆ (0 for inu ign otential energ U 0 for forula U gh 0 for h ( co in or U a a function of,, and Speed at lowet point v 0 for lowet point at / or U equal iniu U 0 for total echanical energ E v / U 0 0 for v U / g[ ( / ] article peed v when i hortet 04 for tenion T becoe zero when i hortet 0 for equation of otion T g in ( 0 for E 0 ( / g[ co in ] 04 for ( tan 05 for 9 / 8 00 for v 4 g / co /6 g 9
art C 4 pt (i 9 (j 4 The peed v H of the particle at it highet point 04 for particle undergoe projectile otion when 0 for angle of projection φ / ( 0 for v H i the -coponent of it velocit at 04 for noting particle doe not trike the urface of the rod 00 for v H 4 g / co( /6 in( / 8 0 44 g The critical value α c of the ratio D/ 04 for particle energ E gd when the weight ha topped 0 for D 0 for E gd v / g[ co in ] 0 for T g in ( 0 for concluding T ut not be negative 06 for an inequalit leading to the deterination of the range of D/ 06 for olving the inequalit to give the range of α D/ 06 for α ( / c 0