Question

Your grandfather clock's pendulum has a length of 0.9930 m.

Answer 1

We will start from the period definition to find the relationship between the perioricity and the length. After finding the relationship between the two lengths and periods we will proceed to calculate the length two with the loss of the period, that is

$T = 2\pi \sqrt{\frac{l}{g}}$

Therefore the period is proportional to the square root of the length

$T \propto l^{1/2}$

Then

$\frac{T_2}{T_1} = \frac{\sqrt{L_2}}{\sqrt{L_1}}$

$(\frac{T_2}{T_1})^2 = \frac{L_2}{L_1}$

$L_2 = (\frac{T_2}{T_1})^2(L_1)$

The period $T_1$ is equivalent to the seconds that a day has, that is 86400 seconds while period two will be the seconds that have one day less the loss of 16 consecutive announced in the statement therefore,

$L_2 = (\frac{86400}{86384})(0.993)$

$L_2 = 0.99336$

The total change in the lenght is

$\Delta L = L_2 - L_1$

$\Delta L = 0.99336 - 0.993$

$\Delta L = 0.00036m = 3.6*10^{-4}m = 360 \mu m$

Therefore the pendulum should be adjust in $360 \mu m$