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