The formula T=2*pi*sqrt[L/g] gives the period of a pendulum of length l feet. The period P is the number of seconds it takes for the pendulum to swing back and forth once. Suppose we want a pendulum to complete three periods in 2 seconds. How long should the pendulum be?
To find the length of the pendulum, we can rearrange the given equation to solve for "L" (length):
T = 2 * π * √[L / g]
Since we want to find the length required for the pendulum to complete three periods in 2 seconds, we can substitute the given values into the equation:
P (period) = 2 seconds
T (desired number of periods) = 3
Plugging in these values, we get:
2 = 2 * π * √[L / g]
Now, we can solve for "L":
Divide both sides of the equation by (2 * π):
2 / (2 * π) = √[L / g]
1 / π = √[L / g]
Square both sides of the equation:
(1 / π)² = L / g
L / g = (1 / π)²
L = (1 / π)² * g
Now, we need to determine the value of "g". The usual approximation for the acceleration due to gravity at the surface of the Earth is about 32 feet per second squared (32 ft/s²).
Substituting the value of "g" into the equation, we get:
L = (1 / π)² * 32
Calculating this expression, we find:
L ≈ 3.28 feet
Therefore, the pendulum should be approximately 3.28 feet long.
To find the length of the pendulum, we can rearrange the formula:
T = 2*pi*sqrt[L/g]
Where:
T = Period of the pendulum (in seconds)
L = Length of the pendulum (in feet)
g = Acceleration due to gravity (approximately 9.8 m/s^2)
Given:
T = 2 seconds
n = 3 (Number of periods)
We can rewrite the formula to solve for L:
L = (T^2 * g) / (4 * pi^2)
Substituting the given values:
L = (2^2 * 9.8) / (4 * pi^2)
L = (4 * 9.8) / (4 * pi^2)
L = 9.8 / (pi^2)
L ≈ 0.987 feet or approximately 0.987 * 12 = 11.85 inches
Therefore, the length of the pendulum should be approximately 0.987 feet or 11.85 inches to complete three periods in 2 seconds.
Solve for L in:
2=2π√(L/g)
π²L/g=1
L=g/π²
Substitute this expression into the formula to see if you do get T=2.