If the hands of a clock coincide every 65 minutes (true time), in 24 hrs the clock will gain (a)

step 1 Hello, so we do a sort of trial error here. So we take each of the answers and if you did it for

If the hands of a clock coincide every 65 minutes (true...

If the hands of a clock coincide every 65 minutes (true time), in $24 \mathrm{hrs}$ the clock will gain
(a) $10 \frac{10}{143}$ mins.
(b) $9 \frac{12}{143}$ mins.
(c) $11 \frac{12}{143}$ mins.
(d) $12 \frac{10}{143} \mathrm{mins}$.

Key Concepts

Clock Gain

Clock gain refers to the amount by which a clock runs fast compared to the actual (true) time. It is measured as the difference between the time indicated by the clock and the correct time over a given period. In contexts like this problem, the gain is deduced by comparing the expected coincidence interval of a correctly functioning clock to the observed interval, allowing one to compute the total accumulated time error over a full day.

Relative Angular Velocity

Relative angular velocity concerns the difference in speeds between the minute and hour hands on a clock. Because the minute hand moves significantly faster than the hour hand, their periodic alignment (coincidence) depends on their relative speeds. Knowing these speeds is crucial for determining the intervals between successive coincidences, which in turn is used to analyze discrepancies in the clock behavior, such as in the calculation of time gain.

Coincidence Interval

The coincidence interval is the time period between successive alignments of the clock’s minute and hour hands. For a standard clock, this interval is defined by the ratio of the angular distances and speeds of the two hands, typically resulting in a period of approximately 65 minutes and a fraction in true time. When a clock deviates from this behavior, the observed coincidence interval can be used as an indicator of the clock's inaccuracy, allowing for computations of how much the clock is gaining or losing time.

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