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A pendulum clock and a digitical clock both are synchronized to beep correct time at temperature 20^(@)C in the mornign on 1st March,2003. At 12:00 noon temperature increases to 40^(@)C and remains constant for three months. Now on Ist June, 2003 at 12:00 noon temperature drops to 10^(@)C and remains constant for a very long duration. Find the date and time on which both the clocks will again be synchronized for a moment. |
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Answer» Solution :As a digital clock (if ideal) always keeps correct time. But on increasing temperatureon 1st March 12:00 noon , the pendulum clock SLOWS doen and start loosing time. We know that tiem lost by a pendulum clock per second is given as `DELTAT=1/2 alpha DeltaT` [If `alpha` is the coefficient of linear expansioin for the material of pendulum] `=1/2 alpha(20)` In three months (March +April+May=92 days) it looses time `Deltat_(92"days")=1/2xxalphaxx20xx92xx86400` One 1st June, 12:00 noon temperature drops to `10^(@)C` which is `10^(@) ` less then the temperatrue at which clock keeps correct time, thus now clock starts gaining time andif after N days it gains exactly the time lost during previous three months, it SHOWS right time again for a moment. Thus time gained by the clock in N days in `deltat_("N days")=1/2 alpha(10)xxNxx86400` We have `deltat_(92"days")` (lost)`=deltat_("N days")` (ganined) or `2/2xxalphaxx90xx92xx86400=1/2xxalphaxx10xxNxx86400` or `N=184` days Thus after 184 days from 1st June 2003, pendulum clock will SHOW correct tiem and bothteh clocks will be in synchronization for a moment and after 184 days means the DATE si 2nd Dec. 2003 and time is 12:00 noon. |
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