Question

If a clock loses 5 seconds per day pendulum

Answer 1

One day is 86400 seconds. If your clock is losing five seconds per day, then it’s only counting out 86395 seconds in that span. In other words, your clock is slow by a factor of:

86395/86400 = 0.99994212963

You need to divide by 0.99994212963^2 (the amount the clock runs slow) not sqrt(x0.99994212963).

Note also that since 0.99994212963=1-0.00005787037, to a very good approximation you need to multiply by 1+2*0.00005787037, or add 2*0.00005787037 the current length.