Question

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

If the clock runs slow and loses 16 s per day, how should you adjust the length of the pendulum?
Note: due to the precise nature of this problem you must treat the constant g as unknown (that is, do not assume it is equal to exactly 9.80 m/s2 ).

Express your answer to two significant figures and include the appropriate units. Enter positive value in case of increasing length of the pendulum and negative value in case of decreasing length of the pendulum.

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 $360 \mu m$