Correct Answer - Option 3 :
\(\frac{1}{{8640}}\) times its original length be shortened
The period of a pendulum is given by
\(T = 2\pi \sqrt {\frac{l}{g}} \)
\(\frac{{dT}}{{dl}} = \frac{{2\pi }}{{\sqrt g }} \times \frac{1}{{2\sqrt l }}\)
\( \Rightarrow \frac{{dT}}{{dl}} = \frac{T}{{2l}}\)
\( \Rightarrow \frac{{dl}}{l} = 2\frac{{dT}}{T}\)
Clock loses 5 seconds per day.
dT = 5 seconds
T = number of seconds in a day = 24 × 60 × 60 = 86400
The alteration required in the length of the pendulum is,
\(\frac{{dl}}{l} = 2 \times \frac{5}{{86400}} = \frac{1}{{8640}}\)
\( \Rightarrow dl = \frac{1}{{8640}} \times l\)
