By what percentage should a pendulum's length be shortened so that the time period decreases from 2t to t seconds.  

Expert Answers

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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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