Marlon F. Sacedon Dept. of Mathematics, Physics, Statistics (DMPS)

1. Generalized Mathematical Formula for Dynamics on One-Dimensional Motion of Two-Object System Formerly ViSCA Marlon F. Sacedon Dept. of Mathematics, Physics, Statistics (DMPS) College of Arts and Sciences, Visayas State University Visca, Baybay City, Leyte, Philippines

2. What is the most important part in the solutions of solving problems in dynamics of motion? Free-body diagram or FBD A vector diagram showing all external forces acting on a body

3. What is the acceleration of the ball if air friction is negligible? FBD ΣFy = ΣF a ΣF mg = ma from Second Law from FBD a = g = 9.8 m/s2 w = mg +y

4. +N -f ΣF a +mgsinθ μk L θ -mgcosθ +y What is the acceleration of the ball if air friction is negligible? ΣFx = ΣF from Second Law from FBD +x +mgsinθ -f = ma mg +mgsinθ -μkN= ma +mgsinθ –μkmgcosθ = ma a = g(sinθ-μkcosθ)

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6. What about this? a +y ΣFB +T +mBgcosø +x -T mB= a ΣFA -μmBg.cosθ +y -mBg.sinθ mA= +mBg +mAg +x Shows complicated FBD!

7. Problem given to students….. Student’s response on FBD. Wrong! Wrong! Wrong! Misconceptions on Free-Body diagram Wrong! Correct! Wrong!

8. This paper presents a generalized mathematical formula that can be used in solving problems of dynamics on one-dimensional motion of two-object system which was derived from the principles of Newton’s Laws of Motion. The formula helps the difficulty of students in solving the problems because the solutions process escapes FBD.

9. Atwood-machine Generalized formula for dynamics on two-object system mB mB mA β mA mA μA mA mB μB mB mA mB mA β θ mB mA mB mA mB mB mA mA mA mB mB

10. Generalized formula for dynamics on two-object system μA μB mB mA β θ Assumptions: • maximum of one pulley and it’s a frictionless. • Air friction is negligible. • motion is due to gravity only. Go to menu • each object moves on a straight line. • object B accelerates down the plane. If calculated a is negative, then it moves up the plane. • Neglect the effects of rotation of pulley & masses. • Negligible weight of cord, and no elongations.

11. Generalized formula for dynamics on two-object system μA μB mB mA β θ Assumptions: • maximum of one pulley and it’s a frictionless. • Air friction is negligible. • motion is due to gravity only. Go to menu • each object moves on a straight line. • object B accelerates down the plane. If calculated a is negative, then it moves up the plane. • Neglect the effects on rotation of pulley & masses. • Negligible weight of cord, and no elongations.

12. Derivation of Formula +y +y +x N mB mA -T ΣFB N μA μB -mAg.sinθ a -mAg.cosθ +x -mBg.cosβ β θ mBg.sinβ +T -μBmBg.cosβ +mAg mBg ΣFB a -μAmAgcosθ Go to menu

13. +y FBD of mA +y FBD of mB +x N -T ΣFB N -mAg.sinθ a -mAg.cosθ +x -mBg.cosβ mBg.sinβ ƩFy =0 ƩFy =0 mBa mAa ΣFx= ΣFx= -μBmBg.cosβ +mAg +T mBg +T–mAgsinθ-μAmAgcosθ=mAa (eq.1) -T+mBgsinβ-μBmBgcosβ=mBa (eq.2) ΣFA Adding equations 1& 2 a -μAmAgcosθ Go to menu +mBgsinβ-μBmBgcosβ –mAgsinθ-μAmAgcosθ =mBa +mAa g[mB(sinβ-μBcosβ –mA(sinθ+μAcosθ )]=a(mB +mA)

14. Common problems on Two-Object system Atwood-machine OR mB mB mA Click buttons to solve mA mA mA mB OR mB mA mB OR mA mB mB mA mA mA mB Next mB Assump

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16. Atwood-machine μB = 0 μA = 0 β= 90o θ = 90o mA mB Generalized Formula: 1 0 1 0 Back to menu

17. Atwood-machine mA mB If mB >mA ,then a>0 If mB <mA ,then a<0 Back to menu ,then a=0 If mB = mA ,then a=g If mA = 0

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19. mB μA = 0 μB mA θ = 90o β Generalized Formula: 1 0 Back to menu

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21. Recall: sin(-A) = -sinA cos(-A) = cosA μA μB = 0 θ = 0o β= 90o mA mB Generalized Formula: 0 ( 1 ) 1 0

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23. μA μB θ = 0o β= 0o Recall: sin(-A) = -sinA cos(-A) = cosA Generalized Formula: OR 1 0 0 1 mA mB mB mA Back to menu

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25. FBD of mA FBD of mB N N +y +y -R +R -mAg -mBg mBa mAa ΣFx= ΣFx= +R-μBmBg=mBa (eq.2) +x +x -R-μAmAg=mAa (eq.1) -μAmAg -μBmBg Adding equations 1& 2 mA mB Back to menu -μBmBg-μAmAg =mBa +mAa

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27. Recall: sin(-A) = -sinA cos(-A) = cosA μA -θ μB β Generalized Formula: (-θ) (-θ) mA Back to menu mB

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29. Recall: sin(-A) = -sinA cos(-A) = cosA μA -θ μB β Generalized Formula: (-θ) (-θ) mA mB OR Back to menu mA mB But : θ=β

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31. +y +y mA mAg.sinβ mBg.sinβ mB μA μB N N FBD of mA FBD of mB -μAmAg.cosβ -μBmBg.cosβ -R ΣFB ΣFA +x +x +R a a -mAg.cosβ -mBg.cosβ Back to menu

32. +y +y mAg.sinβ mBg.sinβ N N FBD of mA FBD of mB -μAmAg.cosβ -μBmBg.cosβ -R ΣFB ΣFA +x +x +R a a -mAg.cosβ -mBg.cosβ Back to menu

33. +y +y mAg.sinβ mBg.sinβ N N FBD of mA FBD of mB -μAmAg.cosβ -μBmBg.cosβ -R ΣFB ΣFA +x +x R a a mBa mAa ΣFx= ΣFx= -R+mAgsinβ-μAmAgcosβ=mAa (eq.1) +R+mBgsinβ-μBmBgcosβ=mBa (eq.2) Adding equations 1& 2 -mBg.cosβ -mAg.cosβ Back to menu +mBgsinβ-μBmBgcosβ +mAgsinβ-μAmAgcosβ =mBa +mAa But : θ=β

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35. Problems that cannot solve by the formula mA mB mA mA mB mB Back to menu

36. Conclusion • The mathematical formula can solve many problems in dynamics on one-dimensional motion of two-object system. • Solutions to the problems becomes shorter and simple. • Escapes totally the constructions of FBD. Back to menu

37. Thanks Email: mfsacedon@yahoo.com Back to menu