## Mikael Group Title The rod is massless. At t = 0 rod is pointing up. It is of length R. rod is rotating without friction around the lower end which is hinged to rotation axis. The mass M is attached at the top end of the rod. The mass is pushed at t=0 with initial velocity . g is gravitational acceleration. All these parameters are given and known. one year ago one year ago

1. Mikael Group Title

2. Mikael Group Title

WHEN WILL THE MASS REACH THE LOWEST POINT ?

3. Mikael Group Title

@mukushla @UnkleRhaukus @experimentex @estudier

4. Mikael Group Title

5. Mikael Group Title

@sauravshakya

6. Mikael Group Title

@Hero @apoorvk @AccessDenied

7. UnkleRhaukus Group Title

find the angular velocity

8. Mikael Group Title

Well it is not constant and it is not explicit in energy - the linear v is there

9. Mikael Group Title

@ganeshie8 @ajprincess

10. Mikael Group Title

@ganeshie8 @ajprincess

11. Mikael Group Title

@ParthKohli @mathslover

12. Mikael Group Title

$0.5mv^2 + mgh = E_0 = constant$

13. Mikael Group Title

CAN ANYONE SOLVE IT AROUND HERE ?

14. Mikael Group Title

@Omniscience ?

15. Mikael Group Title

HINTS ?

16. Mikael Group Title

17. Mikael Group Title

At least some SIGNIFICAN part of the way forward ?

18. Mikael Group Title

19. apoorvk Group Title

Hmm I guess you will just have to find the general term for velocity at any instant, and integrate the term in 'dt' - since energy conservation and angular velocity fundae won't work here.

20. apoorvk Group Title

let me try this out...

21. Mikael Group Title

You ARE 20% forward

22. Mikael Group Title

Hi @ajprincess

23. Mikael Group Title

help

24. Mikael Group Title

Ok , the prize for the sover will be 25 minute-worth field integral from EM that I am ready to do in return !

25. Mikael Group Title

OK the beauty of this situation is that it is VERY common and simple setting. Buuut very unusual answer

26. Mikael Group Title

27. Mikael Group Title

28. Mikael Group Title

@demitris @Directrix can you help ?

29. Mikael Group Title

Hi @sauravshakya

30. Mikael Group Title

@demitris @Directrix can you help ?

31. sauravshakya Group Title

Hi @Mikael ..

32. Mikael Group Title

I AM VERY SORRY I STARTED WITHOUT YOU. Complete please

33. sauravshakya Group Title

Wait a minute.... Let me see what I can do.

34. Mikael Group Title

@Directrix ?

35. Mikael Group Title

36. Mikael Group Title

Hello @Callisto - try to help please

37. Mikael Group Title

Well imagine my sour expression when a nice timid student asks me this question in the middle of a nice sunny day ....

38. Algebraic! Group Title

Why no rotational energy term?

39. Mikael Group Title

Yea , you can use that. Although this is not the only way. Just do it please

40. sauravshakya Group Title

This requires physics too

41. mathslover Group Title

why not to use law of conservation of energy?

42. Mikael Group Title

Calculus and mechanics are essentially one united universal soul

43. Mikael Group Title

Good @mathslover USE it and do the soln. please

44. mathslover Group Title

potential energy + kinetic energy = constant. I cn help after 1 n half hours

45. Mikael Group Title

Actually I have already pointed that in a hint above. But go forward - when is the bottom reached ?

46. Mikael Group Title

@ganeshie8 @UnkleRhaukus help pls

47. Algebraic! Group Title

$\frac{ t ^{2} }{2 } + \frac{ V _{o}}{ R } t =\int\limits_{ }^{ }\int\limits_{ }^{ } \frac {-R }{ g \cos \theta }$ 0..pi

48. Mikael Group Title

A) What is the integration variable ? B) Why (suddenly ) double int ???? c) What is the reason you write this so categorically ?

49. Algebraic! Group Title

A) g, of course B) equation of motion C) integral is tough, not bothering

50. Algebraic! Group Title

theta is the angle the rod makes with the vertical 0...pi

51. Mikael Group Title

g CANNOT CHANGE - it is CONSTANT !

52. Algebraic! Group Title

it's theta, why would you ask?

53. Algebraic! Group Title

would it be g or R?

54. Mikael Group Title

Please Show the FULL argument. Double integral seems, on the face of it, out of place in this problem

55. Mikael Group Title

56. Algebraic! Group Title

argument from torque = -mgRcos(theta) I=mR^2 d^2/ dt^2 (theta) =- gRcos(theta)

57. Mikael Group Title

$E_0 = 2mgR +0.5 mv^2$

58. Mikael Group Title

Giving the solution soon in 3-4 minutes

59. Mikael Group Title

E = CONSTANT

60. Coolsector Group Title

h for you is like the y axis ?

61. Mikael Group Title

yes

62. Mikael Group Title

OK here goes the solution

I have concluded this problem may be above my pay-grade.

64. Mikael Group Title

$t(h) = \int\limits_{h=2R}^{h=0} \frac{dl}{v(h)} dh$

65. Mikael Group Title

Now $dl = \frac{dh}{\cos \theta}$

66. Mikael Group Title

Forgot to say - thta is the angle with the horizon of the rod.The relation dh to dl is a geometric fact that follows from$h = Sin \theta ====> dh = dl * Cos \theta$

67. Mikael Group Title

sorry h = R*Sin theta

68. Mikael Group Title

Thus gathering all the above observations until now we obtain the following integral

69. Mikael Group Title

$\int\limits_{h=2R}^{h=0} dh \frac{1}{\sqrt{1-\frac{h^2}{R^2}}*\sqrt{\frac{2E_0}{m} -2gh}}$

70. Mikael Group Title

Here the 1-st square root in the denominator is simply 1/Cos theta

71. Mikael Group Title

The second square root is the velocity as a function of h - of course derived from the law of conservation of energy stated above

72. Coolsector Group Title

can you please explain how h = rsintheta means dh = dl costheta

73. Mikael Group Title

Now by rescaling the distance by R and taking one convenient constant (from the denominator roots) out of the integral we simplify it all to

74. mathslover Group Title

I did this

75. Mikael Group Title

|dw:1347813111455:dw|

76. mathslover Group Title

$\large{t = \frac{-1+\sqrt{1+8\pi R}}{2v_0}}$

77. Coolsector Group Title

thanks

78. mathslover Group Title

did that make sense? @Coolsector and @Mikael

79. Mikael Group Title

@mathslover I would be amazed and thankful for COMPLETE explanation why do you claim the first equation - that the angle is a simple quadratic function of time. As you know the angular acceleration IS NOT CONSTANT HERE

80. Coolsector Group Title

i dont understand why alpha is what you said.. but maybe its just me

81. Coolsector Group Title

yes .. same problem here

82. Mikael Group Title

I think @mathslover is NOT correct in HIS FIRST STATEMENT on that page on which all is based upon

83. Mikael Group Title

So we proceed

84. Algebraic! Group Title

there's no g in your soln. @mathslover

85. mathslover Group Title

Oh I made it for horizontal plane... vertical plane hmn... well guys wait

86. Mikael Group Title

$\int\limits_{\xi_0}^{0} d \xi \frac{1}{\sqrt{(1-\xi^2)(A- \xi)}}$

87. mathslover Group Title

Mikael whts ur age? grade? so that I can understand the level of this

88. Mikael Group Title

HERE Xi is rescaled variable h, and A is some constant containing fractions of E_0 and g. Outside of the integral is of course a Multiplier (not written) which is physically crucial and mathematically trivial

89. mathslover Group Title

I think this is $$\large{\textbf{above}}$$ my level sorry.... I am in 9th grade and did my best.... sorry

90. Mikael Group Title

This is the level of a good Physical-Course on analytival mechanics in Cambridge/Berkeley/MIT 1-st or second year. Also Ms. @Algebraic! is probably generally pointed to a more Engineering diection of solution - using torques and angular accelerations. Aand Mr. @radar can I send you a message unrelated to math but related to this site ?

91. mathslover Group Title

Sorry.... above my level...

92. Algebraic! Group Title

Ms. I like that snark.

93. Mikael Group Title

I disagree with you @mathslover - you saw it is almost school level

94. mathslover Group Title

What do you mean mikael? well leave ti whatever do you mean........ continue your work..

95. mathslover Group Title

*it

96. Mikael Group Title

Well 1) This is all - please help me by some Tables / Wolframalpha to find what type of Integral this belongs to 2) I suspect it is elliptic Integral

97. Mikael Group Title

Aand please appreciate the effort by medals - you are welcome. The question itself was supplied by Mr. @henpen here http://openstudy.com/study#/updates/50549e72e4b0a91cdf445d5b

98. Mikael Group Title

Here is the answer as supplied by Wolframalph

99. Mikael Group Title

Sorry here in complete - as I thought elliptic integral

100. Mikael Group Title

Can see the soln @mukushla

101. mukushla Group Title

i just came here :)

102. Mikael Group Title

@mukushla The original challenge was for 2 sets of people 1) You, @sauravshakya, and other olympics-math types 2) Physicists So try solving without any reading the soln

103. Mikael Group Title

So I guesses right - about IMO participation ?

104. mukushla Group Title

sorry lost connection Mr.Mikael :) man im a chemical engineer :) what is the math part of question ?

105. Mikael Group Title

You need to find how long it takes for a spherical bead of polymer to slied without friction though a perfectly circular half circular tube to the sovent chamber if initially the small bead has v_0 velocity and starts from the top of the tube

106. Mikael Group Title

solvent chamber (i meant)

107. mukushla Group Title

out of my knowledge :)

108. Mikael Group Title

You should use the simple mechanical conservation of energy and definition of velocity. Some geometry too

109. Mikael Group Title

Well - see the solution and dissolve your sorrows...

110. sauravshakya Group Title

Actually this question is also out of my reach.

111. sauravshakya Group Title

Is this the question from a high school level?