a pendulum-driven clock gains 5.00 s/day, what fractional change
in pendulum length must be made for it to keep perfect time?
1.22
To find the fractional change in pendulum length needed for the clock to keep perfect time, we need to understand the relationship between the period of a pendulum and its length.
The period of a simple pendulum is directly proportional to the square root of its length. Mathematically, it can be represented as:
T = 2π√(L/g)
Where:
T is the period of the pendulum,
L is the length of the pendulum, and
g is the acceleration due to gravity.
We can rearrange this equation to solve for L:
L = (T/(2π))^2 * g
Now, let's analyze the given information. The clock gains 5.00 seconds per day, which means it gains (5.00/24) = 0.2083 seconds per hour. Let's call this gain ΔT.
To find the fractional change in pendulum length, we will use the formula:
ΔL/L = ΔT/T
Substituting the values:
ΔL/L = (0.2083/3600) / (24*60*60)
Simplifying the expression gives us the fractional change in pendulum length needed for the clock to keep perfect time.