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Question Number 19223 by Tinkutara last updated on 07/Aug/17

A particle P is sliding down a frictionless  hemispherical bowl. It passes the point  A at t = 0. At this instant of time, the  horizontal component of its velocity is  v. A bead Q of the same mass as P is  ejected from A at t = 0 along the  horizontal direction, with the speed v.  Friction between the bead and the  string may be neglected. Let t_P  and t_Q   be the respective times taken by P and  Q to reach the point B. Then  (a) t_P  < t_Q   (b) t_P  = t_Q   (c) t_P  > t_Q   (d) (t_P /t_Q ) = ((length of at arc ACB)/(length of chord AB))

$$\mathrm{A}\:\mathrm{particle}\:{P}\:\mathrm{is}\:\mathrm{sliding}\:\mathrm{down}\:\mathrm{a}\:\mathrm{frictionless} \\ $$$$\mathrm{hemispherical}\:\mathrm{bowl}.\:\mathrm{It}\:\mathrm{passes}\:\mathrm{the}\:\mathrm{point} \\ $$$${A}\:\mathrm{at}\:{t}\:=\:\mathrm{0}.\:\mathrm{At}\:\mathrm{this}\:\mathrm{instant}\:\mathrm{of}\:\mathrm{time},\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{component}\:\mathrm{of}\:\mathrm{its}\:\mathrm{velocity}\:\mathrm{is} \\ $$$${v}.\:\mathrm{A}\:\mathrm{bead}\:{Q}\:\mathrm{of}\:\mathrm{the}\:\mathrm{same}\:\mathrm{mass}\:\mathrm{as}\:{P}\:\mathrm{is} \\ $$$$\mathrm{ejected}\:\mathrm{from}\:{A}\:\mathrm{at}\:{t}\:=\:\mathrm{0}\:\mathrm{along}\:\mathrm{the} \\ $$$$\mathrm{horizontal}\:\mathrm{direction},\:\mathrm{with}\:\mathrm{the}\:\mathrm{speed}\:{v}. \\ $$$$\mathrm{Friction}\:\mathrm{between}\:\mathrm{the}\:\mathrm{bead}\:\mathrm{and}\:\mathrm{the} \\ $$$$\mathrm{string}\:\mathrm{may}\:\mathrm{be}\:\mathrm{neglected}.\:\mathrm{Let}\:{t}_{{P}} \:\mathrm{and}\:{t}_{{Q}} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{respective}\:\mathrm{times}\:\mathrm{taken}\:\mathrm{by}\:{P}\:\mathrm{and} \\ $$$${Q}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{point}\:{B}.\:\mathrm{Then} \\ $$$$\left({a}\right)\:{t}_{{P}} \:<\:{t}_{{Q}} \\ $$$$\left({b}\right)\:{t}_{{P}} \:=\:{t}_{{Q}} \\ $$$$\left({c}\right)\:{t}_{{P}} \:>\:{t}_{{Q}} \\ $$$$\left({d}\right)\:\frac{{t}_{{P}} }{{t}_{{Q}} }\:=\:\frac{\mathrm{length}\:\mathrm{of}\:\mathrm{at}\:\mathrm{arc}\:{ACB}}{\mathrm{length}\:\mathrm{of}\:\mathrm{chord}\:{AB}} \\ $$

Commented by Tinkutara last updated on 07/Aug/17

Commented by Tinkutara last updated on 08/Aug/17

Help.

$$\mathrm{Help}. \\ $$

Commented by ajfour last updated on 08/Aug/17

∫(dx/(√(a+cos x))) = ?  ∫_0 ^(  t_P ) dt_P =∫_(−α) ^(  α) ((Rdθ)/(√(v_0 ^2 +2gR(cos θ−cos α))))

$$\int\frac{\mathrm{dx}}{\sqrt{\mathrm{a}+\mathrm{cos}\:\mathrm{x}}}\:=\:? \\ $$$$\int_{\mathrm{0}} ^{\:\:\mathrm{t}_{\mathrm{P}} } \mathrm{dt}_{\mathrm{P}} =\int_{−\alpha} ^{\:\:\alpha} \frac{\mathrm{Rd}\theta}{\sqrt{\mathrm{v}_{\mathrm{0}} ^{\mathrm{2}} +\mathrm{2gR}\left(\mathrm{cos}\:\theta−\mathrm{cos}\:\alpha\right)}} \\ $$

Commented by Tinkutara last updated on 08/Aug/17

This integral is undefined except for a  = 1 as I put this expression in Geogebra  by taking many values.

$$\mathrm{This}\:\mathrm{integral}\:\mathrm{is}\:\mathrm{undefined}\:\mathrm{except}\:\mathrm{for}\:\mathrm{a} \\ $$$$=\:\mathrm{1}\:\mathrm{as}\:\mathrm{I}\:\mathrm{put}\:\mathrm{this}\:\mathrm{expression}\:\mathrm{in}\:\mathrm{Geogebra} \\ $$$$\mathrm{by}\:\mathrm{taking}\:\mathrm{many}\:\mathrm{values}. \\ $$

Commented by Tinkutara last updated on 08/Aug/17

So t_P  can′t be evaluated?

$$\mathrm{So}\:{t}_{{P}} \:\mathrm{can}'\mathrm{t}\:\mathrm{be}\:\mathrm{evaluated}? \\ $$

Commented by ajfour last updated on 08/Aug/17

I cannot.

$$\mathrm{I}\:\mathrm{cannot}. \\ $$

Commented by ajfour last updated on 08/Aug/17

i came across it in trying to    evaluate t_P  .

$$\mathrm{i}\:\mathrm{came}\:\mathrm{across}\:\mathrm{it}\:\mathrm{in}\:\mathrm{trying}\:\mathrm{to}\:\: \\ $$$$\mathrm{evaluate}\:\mathrm{t}_{\mathrm{P}} \:. \\ $$

Answered by ajfour last updated on 08/Aug/17

(a) t_P < t_Q

$$\left(\mathrm{a}\right)\:\mathrm{t}_{\mathrm{P}} <\:\mathrm{t}_{\mathrm{Q}} \\ $$

Commented by Tinkutara last updated on 08/Aug/17

Explain.

$$\mathrm{Explain}. \\ $$

Commented by ajfour last updated on 08/Aug/17

Commented by ajfour last updated on 08/Aug/17

v>v_0   and  cos θ > cos α  v_x =vcos θ    u=v_0 cos α  ⇒  v_x >v_0 cos α (=u) .

$$\mathrm{v}>\mathrm{v}_{\mathrm{0}} \:\:\mathrm{and}\:\:\mathrm{cos}\:\theta\:>\:\mathrm{cos}\:\alpha \\ $$$$\mathrm{v}_{\mathrm{x}} =\mathrm{vcos}\:\theta\:\:\:\:\mathrm{u}=\mathrm{v}_{\mathrm{0}} \mathrm{cos}\:\alpha \\ $$$$\Rightarrow\:\:\mathrm{v}_{\mathrm{x}} >\mathrm{v}_{\mathrm{0}} \mathrm{cos}\:\alpha\:\left(=\mathrm{u}\right)\:. \\ $$

Commented by Tinkutara last updated on 08/Aug/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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