The fractional change in pendulum length required to keep perfect time is approximately 0.004254%. This means the pendulum needs to be lengthened by about 0.004254% to achieve the desired period adjustment.
To solve this problem, we'll use the following steps:
Determine the necessary period change:
The clock gains 7.35 seconds per day, which means its period is shorter than it should be.
A day has 86400 seconds, so the clock's period needs to increase by 7.35 seconds / 86400 seconds = 0.00008507575%.
Relate period change to length change:
The period of a pendulum is directly proportional to the square root of its length.
Mathematically, this relationship can be expressed as: T ∝ √L, where T is the period and L is the length.
To achieve the necessary period change, we need to adjust the length accordingly.
Calculate the fractional change in length:
Since T ∝ √L, we can write: (T + ΔT) / T = √(L + ΔL) / √L, where ΔT and ΔL represent the changes in period and length, respectively.
Squaring both sides and simplifying, we get: (ΔL / L) = [(1 + ΔT / T)^2 - 1]
Plugging in the values for ΔT / T (0.00008507575) and solving, we find: ΔL / L ≈ 0.00004254 * 100% = 0.004254%.
Therefore, the fractional change in pendulum length required to keep perfect time is approximately 0.004254%. This means the pendulum needs to be lengthened by about 0.004254% to achieve the desired period adjustment.
The probable question is in the image attached.