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

=>
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.