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

Answered by BHOOPENDRA last updated on 10/Sep/24

Second hand T_s =60 sec  minute hand T_m =3600 sec  angular velocity  ω=((2π)/T) (where T is time period)  ω_s =((2π)/(60)) rad/sec (for second hand)  ω_m =((2π)/(3600))rad/sec (for minute hand)  Angle of the hand   For the second hand:  θ_s =ω_s .t=((2π)/(60)).t=(π/(30)) . t  For the minute hand:  θ_m =ω_m .t=((2π)/(3600)).t=(π/(1800)).t  for the hand to coincide,their angle  must differ by an integer multiple of  2π  θ_s −θ_m =2πn  (π/(30)).t−(π/(1800)).t=2πn  t=((3600n)/(59))  Number of coincide  interval is 7200 sec  ((3600n)/(59))≤7200  n=118  Note−The first coincidence at 2:01pm  is exclude  The coincidence start just after 2:01&  continue up to just before 4:01pm  therefore ,there are 118 coincidence  during this period.

$${Second}\:{hand}\:{T}_{{s}} =\mathrm{60}\:{sec} \\ $$$${minute}\:{hand}\:{T}_{{m}} =\mathrm{3600}\:{sec} \\ $$$${angular}\:{velocity} \\ $$$$\omega=\frac{\mathrm{2}\pi}{{T}}\:\left({where}\:{T}\:{is}\:{time}\:{period}\right) \\ $$$$\omega_{{s}} =\frac{\mathrm{2}\pi}{\mathrm{60}}\:{rad}/{sec}\:\left({for}\:{second}\:{hand}\right) \\ $$$$\omega_{{m}} =\frac{\mathrm{2}\pi}{\mathrm{3600}}{rad}/{sec}\:\left({for}\:{minute}\:{hand}\right) \\ $$$${Angle}\:{of}\:{the}\:{hand}\: \\ $$$${For}\:{the}\:{second}\:{hand}: \\ $$$$\theta_{{s}} =\omega_{{s}} .{t}=\frac{\mathrm{2}\pi}{\mathrm{60}}.{t}=\frac{\pi}{\mathrm{30}}\:.\:{t} \\ $$$${For}\:{the}\:{minute}\:{hand}: \\ $$$$\theta_{{m}} =\omega_{{m}} .{t}=\frac{\mathrm{2}\pi}{\mathrm{3600}}.{t}=\frac{\pi}{\mathrm{1800}}.{t} \\ $$$${for}\:{the}\:{hand}\:{to}\:{coincide},{their}\:{angle} \\ $$$${must}\:{differ}\:{by}\:{an}\:{integer}\:{multiple}\:{of} \\ $$$$\mathrm{2}\pi \\ $$$$\theta_{{s}} −\theta_{{m}} =\mathrm{2}\pi{n} \\ $$$$\frac{\pi}{\mathrm{30}}.{t}−\frac{\pi}{\mathrm{1800}}.{t}=\mathrm{2}\pi{n} \\ $$$${t}=\frac{\mathrm{3600}{n}}{\mathrm{59}} \\ $$$${Number}\:{of}\:{coincide} \\ $$$${interval}\:{is}\:\mathrm{7200}\:{sec} \\ $$$$\frac{\mathrm{3600}{n}}{\mathrm{59}}\leqslant\mathrm{7200} \\ $$$${n}=\mathrm{118} \\ $$$${Note}−{The}\:{first}\:{coincidence}\:{at}\:\mathrm{2}:\mathrm{01}{pm} \\ $$$${is}\:{exclude} \\ $$$${The}\:{coincidence}\:{start}\:{just}\:{after}\:\mathrm{2}:\mathrm{01\&} \\ $$$${continue}\:{up}\:{to}\:{just}\:{before}\:\mathrm{4}:\mathrm{01}{pm} \\ $$$${therefore}\:,{there}\:{are}\:\mathrm{118}\:{coincidence} \\ $$$${during}\:{this}\:{period}. \\ $$

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

at h:0:0 the minute hand and second  hand coincide.  say at h:m:s they coincide again.  0≤m≤59  0≤s<60  6(m+(s/(60)))=6s ⇒m=((59s)/(60)) ⇒s=((60m)/(59))  ⇒0≤s=((60m)/(59))<60 ⇒0≤m≤58  m=0,1,...,58 ⇒59 times in a hour  2:01 ⇒58  3:00 ⇒59  4:01 ⇒1  Σ: 118 times ✓

$${at}\:{h}:\mathrm{0}:\mathrm{0}\:{the}\:{minute}\:{hand}\:{and}\:{second} \\ $$$${hand}\:{coincide}. \\ $$$${say}\:{at}\:{h}:{m}:{s}\:{they}\:{coincide}\:{again}. \\ $$$$\mathrm{0}\leqslant{m}\leqslant\mathrm{59} \\ $$$$\mathrm{0}\leqslant{s}<\mathrm{60} \\ $$$$\mathrm{6}\left({m}+\frac{{s}}{\mathrm{60}}\right)=\mathrm{6}{s}\:\Rightarrow{m}=\frac{\mathrm{59}{s}}{\mathrm{60}}\:\Rightarrow{s}=\frac{\mathrm{60}{m}}{\mathrm{59}} \\ $$$$\Rightarrow\mathrm{0}\leqslant{s}=\frac{\mathrm{60}{m}}{\mathrm{59}}<\mathrm{60}\:\Rightarrow\mathrm{0}\leqslant{m}\leqslant\mathrm{58} \\ $$$${m}=\mathrm{0},\mathrm{1},...,\mathrm{58}\:\Rightarrow\mathrm{59}\:{times}\:{in}\:{a}\:{hour} \\ $$$$\mathrm{2}:\mathrm{01}\:\Rightarrow\mathrm{58} \\ $$$$\mathrm{3}:\mathrm{00}\:\Rightarrow\mathrm{59} \\ $$$$\mathrm{4}:\mathrm{01}\:\Rightarrow\mathrm{1} \\ $$$$\Sigma:\:\mathrm{118}\:{times}\:\checkmark \\ $$

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

for a mechanic clock the minute  hand coincides with the second hand  at following moments in a hour:  h:00:00  h:01:01  h:02:02  ...  h:59:59  totally 60 times in a hours.  in the period 2:01 till 4:01 they  coincide 120 times.

$${for}\:{a}\:{mechanic}\:{clock}\:{the}\:{minute} \\ $$$${hand}\:{coincides}\:{with}\:{the}\:{second}\:{hand} \\ $$$${at}\:{following}\:{moments}\:{in}\:{a}\:{hour}: \\ $$$${h}:\mathrm{00}:\mathrm{00} \\ $$$${h}:\mathrm{01}:\mathrm{01} \\ $$$${h}:\mathrm{02}:\mathrm{02} \\ $$$$... \\ $$$${h}:\mathrm{59}:\mathrm{59} \\ $$$${totally}\:\mathrm{60}\:{times}\:{in}\:{a}\:{hours}. \\ $$$${in}\:{the}\:{period}\:\mathrm{2}:\mathrm{01}\:{till}\:\mathrm{4}:\mathrm{01}\:{they} \\ $$$${coincide}\:\mathrm{120}\:{times}. \\ $$

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