# By how much should the length of a pendulum be shortened if with its present length it loses half a minute per day?

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The time period of a pendulum is given by 2*pi*sqrt (L/g), where L is the length of the pendulum.

In the problem, the pendulum loses half a minute per day. This implies that the length of the pendulum is longer than what it should be making the time period not equal to 2 sec, but instead making it:

2 + (1/2)/ (24*60*60)

=> 2 + .005787037

=> 2.000005787

For the longer time period of 2.000005787, the length of the pendulum is [(2.000005787)^2*g/ (4*pi^2)] which is equal to 0.992953345 m

As we require the time period of the pendulum to be equal to 2 seconds, its length should be: [2^2*g/(4*pi^2)] = .992947599 m

Therefore the length of the pendulum has to be shortened by:

2.000005787^2*g/(4*pi^2) - g/pi^2

=> .005746236*10^-3 m

=> 0.5746 cm

=> 5.746 mm

**The length of the pendulum has to be shortened by approximately 5.746 mm.**