The length of time, t (in seconds), it takes the pendulum of a clock to swing through one complete cycle is a function of the length of the pendulum in feet. Defined by:
t = f(L) = 2pi sqrt L/32
A.Rewrite the formula using fractional exponents.
B.Determine the length of the cycle in time if the pendulum is 4 feet long.
good grief, g = 32 ft/s^2, old text :)
T = 2 pi (L/32)^(1/2)
T = 2 pi (4/32)^.5
= 2.22 seconds
A. To rewrite the formula using fractional exponents, we can express the square root as an exponent of 1/2. Therefore, the formula can be rewritten as:
t = f(L) = 2π(L/32)^(1/2)
B. To determine the length of the cycle in time if the pendulum is 4 feet long, we can substitute L = 4 into the formula:
t = 2π(4/32)^(1/2)
Simplifying the expression inside the square root:
t = 2π(1/8)^(1/2)
Taking the square root of 1/8:
t = 2π(1/√8)
Simplifying the denominator:
t = 2π(1/√(4*2))
t = 2π(1/(2√2))
Since √2 is irrational, we cannot simplify it any further. Therefore, the length of the cycle in time for a 4-foot long pendulum is:
t = 2π/(2√2)