Final answer:
To correct a pendulum-driven clock gaining 5.00 seconds per day, we calculate the needed fractional change in length using the period formula. The correct approximation of the change is 0.0058%, which is closest to option (b) 0.025%, but it is not an exact match to the listed options.
Step-by-step explanation:
To determine the fractional change in pendulum length that must be made for a pendulum-driven clock that gains 5.00 seconds per day to keep perfect time, we can use the pendulum period formula:
T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
The period is proportional to the square root of the length of the pendulum. If the clock is gaining time, then its period is too short, and we need to increase the length of the pendulum slightly. For a small fractional change in length ΔL/L, the fractional change in the period ΔT/T is approximately 2(ΔL/L).
Over one day (86,400 seconds), the clock gains 5 seconds, so the fractional change in period is 5/86400. Therefore, 2(ΔL/L) = 5/86400, which gives us ΔL/L = 5/(2*86400). When this calculation is performed, the result is approximately 0.000058, or 0.0058%. So compared to the given options, the closest answer is (b) 0.025%, though it is still not quite the precise figure obtained by the calculation.