Chapter 14 Exercise 14A
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1 hapte 1 Eecise 1 Q. 1. d Q.. Mass d (i), d d di d I d I [ ] () 0 I 8 6 I (ii) k I QED k I O Q.. d d d d d I Mass ea p ( 1 ) 8p d d p d 8p d S, di d di d I 1 d [ ] 16 1 d 16 [ 81 1 ] I 5 QED Mass f annuus: d (i) di d, I d d d_ d d d [ ] d d[ d 8 + d 8 ] d d I d 1 (ii) k I d 1 k d QED d d d d d dd d d Q.. 6 d di d But d d 6 d I 5 d 6 [ 5 ] 18 d 18 [ 15 + ] 7 6 1
2 Q. 5. I _ d d d k [ ( ] ) ( ) ( ) k d d ( + ) QED Nting that ( )( + ) k d k _ d Q. 7. d nw di d I 0 d I () [ ] 0 y 8 I QED k I k Nting, hee ass Mass, 8 7y _ 8 p ( 9y y ) 8 p(8)y () py Q. 6. d d d annua ass p d y d d Nw di d Mass ea p( ) d d p( p d ) d I 7y y y d 7y (y ) [ ] y_ y [ y [ _,01 16 y y 16 ],00y 16 ] nw, by Definitin di d I 50y QED di _ ( ) d
3 Q. 8. p Q.. (i) p d d, d p d p d d d p p(d) _ d p nw di d I p d p Eecise 1B p p [ ] (p) p p [ p + p ] p QED (ii) d: I X : Disc: I : 1 aae es: (I X I + ) I X + () _ I X 19 I Tta I d + I Disc _ 19 + (57 + 8) 6 I 65 6 Q. 1. (i) I 1 ()() (ii) I ()() 1 (iii) I c + d + () 7 Q.. (a) (i) I 1 () 1 (ii) I B 1 ()() p c aae es: (I X I + ) Disc: I X + I X t Mass: I X () I X (iii) I I + I B (by pependicua ais thee) 5 (iv) I c + d 5 + () 17 (iii) I Tta + _ 11 B 6 B (b) Ony (iv) and by ()() 1
4 F ectange: d: I _ I : (), I BB () I X Syste: I Tta I d + I Laina ependicua es Thee: I X I + I BB I X 9 + _ 10 Q.. p _ q c (iv) 6 d: ()() 16 t ass: 5 1 I X () 16 Disc: (I X I + ) Disc: I c + d 1 ()( ) + ()(5) 51 _ () + () 19 d: Q. 5. (i) Tta B I X (6) (v) 8 I Tta B By pependicua ais thee I + I B I (ii) But since I I B I 1 I I 1 ( 1 ) 1 (by syety) B Q. 6. Laina: I I BB es I I + I BB I es (I X I + ) I + () _ I X 9 Standad sitin gh I w 1 gh + 1 I w g(1) + 1 I(0) g(0) + 1 I w I w g w g w g
5 Q. 7. (i) q d: I () 16 Disc: I I + d 1 + (51) 51 Tta (ii) gh+ gh + 1 Iw gh + gh + 1 Iw d Disc Syste d Disc Syste g(7) + g(10) + 1 I(0) g() + g(0) + 1 Iw v w 5 168g 185 Q. 8. The ectangua aina: Iw 8g w 8 g w 168g 185 Standad sitin B I 1 () I B 1 ( ) I I + I B I I + d ( ) 51 5
6 The pint ass: (6) 6 The syste gh + gh + 1 Iw gh + gh + 1 Iw Laina int ass Syste Laina int ass Syste g (6) + g (6) + 1 (0) g() + g(0) + 1 Iw Iw 18g 195 w 18g w 7g 195 g 65 Speed f k w 6 g 65 6 k k k sitin 1 sitin Standad sitin Q.9. p y The d: The pint ass: The syste: I () I () I 8 gh + gh + 1 Iw gh + gh + 1 Iw d int ass Syste d int ass Syste ()g() + g( ) + 1 I ( g ) ()g + g(0) + 1 Iw 9g + g + 1 (8 ) ( g ) g + 1 (8 )w w 1g w 7g y sitin 1 sitin Standad y sitin Q. 10. (i) 6 Laina: Q I Q () I yy d: I () es: I O I + I Q 5 1 6
7 (ii) Then (I I O + ), ( es Thee) Syste: I 5 + (8) I 197 I Tta I d + I Laina sitin 1 Eecise 1 Q N 0 g adius sin 0 (i) Gain in K.E. Lss in.e. 1 v + 1 Iw gh F w v 1 v + 1 ( 1 ) ( v ) g(10 sin 0 ) 1 v + 1 v 60g v 60g v 80g v 80g 8 /s 5 Standad sitin sitin Enegy nseved: Q.. (i) (ii) v u + as a v u s 8 0 (10) v u + at t v a u 8 ( ) N g cs q s..e. 1 + K.E. 1.E. + K.E. g(11) + g(16) q F g(5) + g(0) + 1 Iw 5g 1 Iw But I 09 w 108g 09 w 108g 09 5 q g Gain in K.E. Lss in.e. 1 v + 1 Iw gh 1 v + 1 ( 1 ) ( v ) g(s sin q) 1 v + 1 v gs ( 5 ) utipy by 0 7
8 10v + 5v 1gs 15v 1gs v gs 5 v u + as gs as g 10a a 5 g (ii) On the pint f sipping g sin q F a Q.. Fces: g ( 5 ) g ( 5 ) ( g 1 QED 5 ) Q.. Lking at the fces n the ass: F a g T ( g 5 ) 10g 5T 8g 5T g q T 5 g N T g T T g (i) Gain in K.E. Lss in.e. 1 v + 1 Iw gs 1 v + 1 ( 1 ) ( v ) gs 1 v + 1 v gs v gs Fisty, we find the speed f the ass afte it has faen a distance h. Gain in K.E. Lss in.e. 1 v + 1 Iw gh ass disc ass 1 ()v + ( 1 1 ) ( v ) gh v + 1 v gh 5 v gh v 8gh 5 v u + as a v u s 8gh 5 0 h (ii) v u + as a v u s (iii) F a g T ( g ) g T g T g T 1 g Q. 5. (i) and 1 I 1 ( + 1) 5 v gs v gs _ gs 0 s g 5 g s 8
9 Fces: Q. 6. Fces: F 5 g T S esved: T S g g 5 F g g Gain in K.E. Lss in.e. 1 v + 1 Iw gh 1 v + ( 1 5 )( v ) g(s sin 5 ) 1 v + ( 1 5 )( v ) g ( s ) 16 utipy by 8v + 5v 16gs 1v 16gs v 16gs 1 v u + as 16gs a v u 1 0 s s 8g 1 /s (ii) F a ( assue annuus is n the pint f sipping) g ( g ) ( 8g 1 ) 1 utipy by g This is the east vaue f that wi pevent sipping 5 1 Q. 7. (i) Gain in K.E. Lss in.e. 1 v + 1 v + 1 Iw gh gh ass ass disc ass ass 1 v + 1 ()v + ( 1 1 ) ( v ) gh gh utipy by v + 8v + v 1gh 11v 1gh v 1 11 gh v u + as a v u 1 11 gh 0 6 s h 11 g (ii) ass: F a g T ( 6 11 g ) g 11T g ass: 11T 0g T 0 11 g F a S g ( 6 11 g ) 11S 11g 6g N 0 11S 17g S g adius F 0 g 9
10 Q. 8. Gain in K.E. Lss in.e. 1 v + 1 Iw gh 1 v + 1 ( )( v ) g(s sin 0 ) assuing it s a distance s dwnhi 1 v + 1 v gs v gs v u + as a v u gs _ 0 1 s s g y p ( ) di y d y d p y (y) dy I dy p p( ) y dy ( ) [ y ] ( ) ( ) _ (i) ( )( + ) ( ) I ( + ) QED I ( + ) NZL: ΣL Iq.. but p: g ( + ) q.. (ii) Gain in K.E. Lss in.e. 1 v + 1 Iw gh 1 v + 1 [ 1 ( + ) ] ( v ) gh utipy by v + ( + )v gh v ( + + ) gh v ( + ) gh v gh + v u + as a v u s _ gh + 0 h g + (iii) F a Eecise 1D g T ( g + ) T g ( 1 + ) g ( + + ) g ( + + ) 10 T O g I ( + ) + Q. 1. d: I Length: ass: F the pund enduu: I T p gh Hee h : T p g T p g
11 Q.. Disc: I O I + I I T p gh p g p g O Q.. The d: y I The syste: I 16 The pint ass: I () Fces: esutant: h g y g Q.. g Taking ents abut : g() + g() gh X G X h ass: I XX I (Squae Laina) I XX (Standad Fua) ependicua es: I G I + I yy I G I I G aae es: I I G + Hee S I + ( ) I 8 I T p gh p 8 g p 8 g Q. 5. The ass f the syste is. I T p gh p 16 p ()g( ) p 16 9g 8 p g The d: I The disc: I I + d 1 () + ()() 19 The syste: I
12 T find h: Fces: II es: I I + p g g _ I 15 + (9) 9 esutant: I Tta I d + I Laina h g g() + g() g(h) 87 Find sitin f ente f Gavity G: Q. 6. (i) h 7 The ass f the syste is. I T p gh 61 p ()g( 7 ) p 61 1g X X Ments abut p: () + (9) h h 6 I S T p gh p 87 ()g(6) p 19 16g (ii) Sipe enduu Equivaent: k p g p 19 16g k Q. 7. I I + d d: I () I 1 Laina: I XX ( ) I () es: I I XX + I _ 15 h I T p gh _ 1_ p + g p 7 6g 1_ ( )( ) 0 O 1
13 Q. 8. (a) B (ii) Fces: g g (b) (ii) I I + I B c The d: The aina: I c + d + () 9 The syste: esutant: h g Taking ents abut p: g() + g() gh h The ass f the syste is. I T p gh p 11 ()g() p 11 g Q. 9. (i) sitin 1 q sitin q Standad sitin The disc: I c + d 1 + The pint ass: ()() 8 The syste:
14 gh + gh + 1 Iw gh + gh + 1 Iw Disc int Mass Syste Disc int Mass Syste g() + ()g() + 1 I(0) g() + ()g(0) + ( 1 19 ) w (ii) T find h: Fces: 5g 19 w w 0g 19 esutant: g g g g() + g() g(h) h 5 The ass f the syste is. 19 I T p gh p ()g 5 p 19 10g If this equas p g, then Q. 10. (i) The d: Taking ents abut a: a ()p p The pint ass: I a y The syste p + y (ii) T find h: Fces a y g g esutant b gy + gp gh h y + p The ass f the syste is. I T p gh p + y p g ( y + p ) p p + y g(y + p) But this equas p 0p g p + y y + p 0p 1 a h g b y 0py + 1p 0 (y p)(11y 6p) 0 y p O 6p 11
15 Q. 11. (i) Standad f (ii) S I Tta 8 + ( ) I Tta 16 QED + (16 ) X O I XX I G X (iv) Nte: By syety, cente f gavity f syste is at G h in pat(ii) I T p gh T p 16 5 g T p 8 QED 5g ependicua es: I G I + I yy Eecise 1E I G aae es: Q. 1. c (iii) I O I G + whee I O + ( ) I O 8 D Laina: G B F (ii) I O 8 ds and B: Length ass I I O I +, ( es) I O + ( + ) I T p gh T p ( + ) g dt d [ p 1 1 g + g ] 1 [ [ 1_ dt g d p g ] g + g _ ] 1_ [ g 1 g p [ 1 g + g ] 1 g + 1 g ] 0 f iniu T I O ds and D: + 5 I O I Mid pint + + ( 5 ) _ I O 16 F iniu T Q.. (, ) t Mass: d: I O I O 15
16 es: Q.. I O I + + I Tta I Mass + I d G + + _ 9 + T p [ I gh ] T p 9 + ()g T p [ g + 9g ] 1_, F Min T, dt d 0 dt d p [ g + 9g ] 1 [ 1 g 9g ] 1 g 9g F Min T Disc: I G es: + _ + T p [ I gh ] T p + g Q.. Squae: I I yy es: I G ZI X G X If, we get + T p g p g F iniu T, dt d 0 es: I G + + T p [ I gh ] p + g p [ g + g ] 1 dt d p [ dt d 0 g + g 1 g g ] 1 [ g + 1 g ] F iniu T QED F T p [ g + g ] 1... dt d p [ dt d 0 g + 1 g + g ] 1 [ g F T p + g p g < p g since < g + 1 g ] QED 16
17 Q. 5. Squae Laina: Q. 6. (a) Standad f G I (b) Standad f (c) d: I a Disc: I ( a ) I a a c X a I es: I a + I Tta I d + I Disc I (i) I ( ) I G I ( es) I G es: I I G + I + t. Mass: : I (1 ) : I (1 + ) I Tta I Laina + I t. Mass + + (1 + ) + (1 + + ) + 5 I (ii) T p, By syety cente f gh gavity f syste is at G. T p [ ( + 5 ) 5g ] (iii) T p 5g [ + ] 5 QED d(t ) d p 5g { 1 [ (10) ( + 5 )( 1) ] p 5g [ 5 ] F Miniu T, dt d 0 a + a + (i) I TOTL (a + 1 ) QED (ii) Nte: In this questin the pane f the disc is pependicua t the pane in which the d ves, i.e. as shwn in diaga the disc ves Int the page. I Nw, T p gh We need t find h, (the psitin f cente f gavity) Taking ents abut : (a) + 5h h a + 5 S T p (a + 1 )(5) 5(g)(a + ) T p (a + 1 ) g(a + ) Nw, Miniuu T ccus f (a + 1 ) iniised (a + ) 1 (a + ) [ (a + )() (a + 1 )() ] (a + )() (a + 1 )() a + a a a 0 ( + a)( a) a F iniu T QED 17
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