CHAPTER 18 MOTION IN A CIRCLE

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1 EXERCISE, Pae 6 CHAPTER 8 MOTION IN A CIRCLE. A locomotive travels around a curve of 0 m radius. If te orizontal trust on te outer rail is of te locomotive weit, determine te speed of te locomotive. Te surface tat te rails are on may be assumed to be orizontal and te orizontal force on te inner rail may be assumed to be zero. Centrifual force on outer rail m Hence, r m from wic, v r m /s v m/s m km 3600s 9.9 s 0 m te speed of te locomotive, v km/. If te orizontal trust on te outer rail of Problem is determine its speed. of te locomotive s weit, Centrifual force on outer rail m Hence, r m from wic, v r m /s v m/s 58

2 (7 3.6) km/ te speed of te locomotive, v 5. km/ 3. Wat anle of bankin of te rails of Problem is required for te outer rail to ave a zero value of outward trust? Assume te speed of te locomotive is 5 km/. Anle of bankin, θ v 5 km/ v r km 0 m s km m/s Hence, θ m / s 0m 9.8m / s ( ) anle of bankin, θ Wat anle of bankin of te rails is required for Problem 3, if te speed of te locomotive is 30 km/? Anle of bankin, θ v 30 km/ v r km 0 m s km m/s Hence, θ m / s 0m 9.8m / s (0.0458) anle of bankin, θ

3 EXERCISE 0, Pae 8. A conical pendulum rotates about a orizontal circle at rpm. If te speed of rotation of te mass increases by 5%, ow muc does te mass of te pendulum rise? Anular velocity, ω π n π 0.47 rad/s ω or ω from wic, eit, ω m Wen te speed of rotation rises by 5%, n rpm πn π 05 Hence, ω rad/s ω or ω Hence, ω te new value of eit, m Rise in eit of te pendulum mass old new m 8.3 mm. If te speed of rotation of te mass of Problem decreases by 5%, ow muc does te mass fall? From Problem, m Wen te speed of rotation decreases by 5%, n rpm πn π 95 Hence, ω rad/s 60

4 ω or ω Hence, ω te new value of eit, m Fall in eit of te pendulum mass old new m 9.66 mm 3. A conical pendulum rotates at a orizontal anular velocity of rad/s. If te lent of te strin is 3 m and te pendulum mass is 0.5 k, determine te tension in te strin. Determine also te radius of te turnin circle. Anular velocity, ω rad/s Tension in te strin, T m ω L 0.5 k ( rad/s) 3 m T 3 k m/s However, k m/s N, ence, tension in te strin, T 3 N T m cos θ m from wic, cos θ T 0.5k 9.8m / s 3N Hence, te cone anle, θ cos (0.875) sin θ r, from wic, radius of turnin circle, r L sin θ 3 m sin L.78 m 6

5 EXERCISE 0, Pae 30. A uniform disc of diameter 0. m rotates about a vertical plane at 00 rpm. Te disc as a mass of.5 k attaced at a point on its rim and anoter mass of.5 k at anoter point on its rim, were te anle between te two masses is 90 clockwise. Determine te manitude of te result centrifual force tat acts on te axis of te disc, and its position wit respect to te.5 k mass. Anular velocity, ω π n π rad/s 0. Force, F mω r N 0. Force, F mω r N From te diaram, result, R ( F ) ( F ) + + by Pytaoras N cos θ R and θ cos (0.55) 59º clockwise. If a mass of 4 k is placed on some position on te disc in Problem, determine te position were tis mass must be placed to nullify te unbalanced centrifual force. From Problem, result, R N mω r r from wic, radius, r m 6

6 Hence, radius, r mm at (80º - 59º) anticlockwise to te.5 k mass radius r mm at º anticlockwise to te.5 k mass 3. A stone of mass 0. k is wirled in a vertical circle of m radius by a mass-less strin, so tat te strin just remains taut. Determine te velocity and tension in te strin at (a) te top of te circle, (b) te bottom of te circle, (c) midway between (a) and (b). (a) At te top, tension T 0 N anular velocity, mω r m 9.8 ω 3.3 rad/s r and linear velocity, v ωr 3.3 rad/s m velocity at te top, v 3.3 m/s and te tension 0 N (b) At te bottom, (kinetic enery + potential enery) top (kinetic enery) bottom + m v v

7 and v m/s Tension at te bottom, v + ω + r T m m r m N velocity at te bottom, v 7 m/s and te tension 5.88 N (c) Mid-way, (kinetic enery + potential enery) top (kinetic enery + potential enery) mid way 3 + m + m v v and v m/s Tension mid-way, T3.94 N r velocity mid-way, v m/s and te tension.94 N EXERCISE 03, Pae 30 Answers found from witin te text of te capter, paes 3 to 30. EXERCISE 04, Pae 30. (b). (d) 3. (a) 4. (b) 5. (b) 64

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