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Question Number 84785 by john santu last updated on 16/Mar/20

$$\mathrm{at}\mathrm{what}\mathrm{time}\mathrm{is}\mathrm{the}\mathrm{short}\mathrm{clock}$$$$\mathrm{and}\mathrm{long}\mathrm{hour}\mathrm{hand}\mathrm{form}\mathrm{an}$$$$\mathrm{angle}\mathrm{of}180\mathrm{degrees}?$$

Commented by mr W last updated on 16/Mar/20

$$hourandminutehand?$$

Commented by john santu last updated on 16/Mar/20

$$\mathrm{yes}.$$

Answered by mr W last updated on 16/Mar/20

$$at0:00hbothhandsareatposition$$$$\theta =0.$$$$$$$$attimehhourandmminute,thepositionof$$$$minutehandis$$$${\theta }_m=m$$$$thepositionofhourhandis$$$${\theta }_h=(h+\frac{m}{60})\times 5=5h+\frac{m}{12}$$$$$$$$whenbothhandsformanangleof$$$$180°,itmeans$$$$\mid {\theta }_m-{\theta }_h\mid =30$$$$m-5h-\frac{m}{12}=\pm 30$$$$\frac{11m}{12}-5h=\pm 30$$$$\Rightarrow 11m-60h=\pm 360$$$$$$$$for11m-60h=360$$$$\Rightarrow m=\frac{60(h+6)}{11}⩽60\Rightarrow h⩽5$$$$h=0\Rightarrow time{0}^h{32}^m{43}^s$$$$h=1\Rightarrow time{1}^h{38}^m{10}^s$$$$h=2\Rightarrow time{2}^h{43}^m{38}^s$$$$h=3\Rightarrow time{3}^h{49}^m{05}^s$$$$h=4\Rightarrow time{4}^h{54}^m{32}^s$$$$h=5\Rightarrow time{5}^h{60}^m{00}^s\Rightarrow 6^h{00}^m{00}^s$$$$$$$$for11m-60h=-360$$$$\Rightarrow m=\frac{60(h-6)}{11}⩾0\Rightarrow h⩾6$$$$h=6\Rightarrow time{6}^h{00}^m{00}^s$$$$h=7\Rightarrow time{7}^h{05}^m{27}^s$$$$h=8\Rightarrow time{8}^h{10}^m{54}^s$$$$h=9\Rightarrow time{9}^h{16}^m{21}^s$$$$h=10\Rightarrow time{10}^h{21}^m{49}^s$$$$h=11\Rightarrow time{11}^h{27}^m{16}^s$$$$h=12\Rightarrow time{12}^h{32}^m{43}^s$$

Commented by john santu last updated on 16/Mar/20

$$\mathrm{waw}\mathrm{great}\mathrm{sir}.$$$$\mathrm{i}\mathrm{find}\mathrm{with}\mathrm{my}\mathrm{formula}$$$${\theta }_{\mathrm{h}}=\mathrm{for}\mathrm{hour}$$$${\theta }_{\mathrm{m}}=\mathrm{for}\mathrm{minute}$$$$\mathrm{the}\mathrm{angle}\alpha =\mid 30\times {\theta }_{\mathrm{h}}-\frac{11}{2}\times {\theta }_{\mathrm{m}}{\mid }^{\mathrm{o}}$$$$\mathrm{if}<{180}^{\mathrm{o}}$$$$\mathrm{and}\beta ={180}^{\mathrm{o}}-\alpha ,\mathrm{if}>{180}^{\mathrm{o}}$$$$$$$$$$

Commented by mr W last updated on 16/Mar/20

$$thecalculatedinstantsaretheoretic,$$$$basedontheassumptionthatthehands$$$$movecontinuously.inrealitythe$$$$handsofaclassicmechanicclock$$$${don}^{\prime }tmovecontinuously.forexample,$$$$thesecondhandmovesastepevery$$$$second,theminutehandmovesastep$$$$every60seconds,thehourhandmoves$$$$astepevery12minutes.whenwe$$$$considerthiseffect,{we}^{\prime }llgetvery$$$$differentandinterestingthings.$$$$seeQ17449forexample.$$

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