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Question Number 211455 by BaliramKumar last updated on 09/Sep/24

Answered by BHOOPENDRA last updated on 10/Sep/24

$$Secondhand{T}_s=60sec$$$$minutehand{T}_m=3600sec$$$$angularvelocity$$$$\omega =\frac{2\pi }{T}\left(whereTistimeperiod\right)$$$${\omega }_s=\frac{2\pi }{60}rad/sec\left(forsecondhand\right)$$$${\omega }_m=\frac{2\pi }{3600}rad/sec\left(forminutehand\right)$$$$Angleofthehand$$$$Forthesecondhand:$$$${\theta }_s={\omega }_s.t=\frac{2\pi }{60}.t=\frac{\pi }{30}.t$$$$Fortheminutehand:$$$${\theta }_m={\omega }_m.t=\frac{2\pi }{3600}.t=\frac{\pi }{1800}.t$$$$forthehandtocoincide,theirangle$$$$mustdifferbyanintegermultipleof$$$$2\pi$$$${\theta }_s-{\theta }_m=2\pi n$$$$\frac{\pi }{30}.t-\frac{\pi }{1800}.t=2\pi n$$$$t=\frac{3600n}{59}$$$$Numberofcoincide$$$$intervalis7200sec$$$$\frac{3600n}{59}⩽7200$$$$n=118$$$$Note-Thefirstcoincidenceat2:01pm$$$$isexclude$$$$Thecoincidencestartjustafter2:01\mathrm{\&}$$$$continueuptojustbefore4:01pm$$$$therefore,thereare118coincidence$$$$duringthisperiod.$$

Answered by mr W last updated on 10/Sep/24

$$ath:0:0theminutehandandsecond$$$$handcoincide.$$$$sayath:m:stheycoincideagain.$$$$0⩽m⩽59$$$$0⩽s<60$$$$6(m+\frac{s}{60})=6s\Rightarrow m=\frac{59s}{60}\Rightarrow s=\frac{60m}{59}$$$$\Rightarrow 0⩽s=\frac{60m}{59}<60\Rightarrow 0⩽m⩽58$$$$m=0,1,...,58\Rightarrow 59timesinahour$$$$2:01\Rightarrow 58$$$$3:00\Rightarrow 59$$$$4:01\Rightarrow 1$$$$\mathrm{\Sigma }:118times✓$$

Commented by mr W last updated on 11/Sep/24

$$foramechanicclocktheminute$$$$handcoincideswiththesecondhand$$$$atfollowingmomentsinahour:$$$$h:00:00$$$$h:01:01$$$$h:02:02$$$$...$$$$h:59:59$$$$totally60timesinahours.$$$$intheperiod2:01till4:01they$$$$coincide120times.$$

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