By what percentage should a pendulum's length be shortened so that the time period decreases from 2t to t seconds.
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The period of oscillation of a pendulum is given by the formula T = 2*pi*sqrt( L / g) where, L is the length of the pendulum and g is the acceleration due to gravity.
In the given problem the period of the pendulum is 2t.
=> 2t = 2*pi*sqrt( L / g)
To reduce the time period to t, let the length of the pendulum required be L'.
t = 2*pi*sqrt( L' / g)
=> [2*pi*sqrt( L / g)] / 2
2*pi*sqrt( L' / g) = [2*pi*sqrt( L / g)] / 2
=> sqrt (L'/g) = [sqrt (L/g)]/2
=> 2*sqrt (L'/g) = [sqrt (L/g)]
=> 2*sqrt L' = sqrt L
=> 4L' = L
=> L' = L/4
Therefore the pendulum's length should be reduced by 75%.
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