1=2pi times the square root of (L/9.8)

Period of pendulum = 1 second what is its length ?

1^2 = (2pi)^2 L/9.8

L = 9.8/(2pi)^2

L = .248 meters or 24.8 cm, about 10 inches

The equation you provided is an interesting one. It is actually the equation for the period of a simple pendulum. Let me explain how to derive it.

A simple pendulum consists of a mass (typically referred to as a "bob") attached to a string or rod of length L. When the pendulum is displaced from its resting position and then released, it swings back and forth in an arc. The time it takes for one complete swing (from one extreme point to the other and back) is known as the period.

To derive the equation you mentioned, we need to consider a few concepts from physics, such as gravity and the restoring force of a pendulum.

1. Starting with Newton's second law, we have: F = ma, where F is the net force acting on the pendulum bob, m is its mass, and a is its acceleration.

2. For a simple pendulum, the only force acting on the bob is gravity, which causes it to swing back towards its equilibrium position. This force is given by F = mg, where g is the acceleration due to gravity.

3. The restoring force of the pendulum is proportional to the displacement from the equilibrium position. This force can be described as F = -kx, where k is the spring constant and x is the displacement. In this case, the displacement can be approximated by the arc length of the swing, which is directly proportional to the angle of displacement.

4. Using trigonometry, we can relate the angle of displacement θ to the arc length s: s = Lθ. Here, L is the length of the pendulum.

5. Combining the equations F = ma = -kx and F = mg, and substituting the expression for x, we obtain: ma = -k(Lθ).

6. The restoring force of a pendulum can also be expressed as F = -mLω²sin(θ), where ω is the angular frequency of the pendulum. This equation can be derived using rotational mechanics.

7. Equating the two expressions for the restoring force, we get: mω²Lsin(θ) = -kLθ.

8. Rearranging the equation, we have: ω² = -k/mb.

9. Using the small angle approximation sin(θ) ≈ θ (valid for small angles), and substituting ω = 2πf (where f is the frequency of the pendulum), we get: (2πf)² = -k/mb.

10. Rearranging the equation, we find: f = 1/(2π)√(k/m).

11. Since the period T is the reciprocal of the frequency (T = 1/f), we get: T = 1/(2π)√(m/k).

12. Simplifying further, we have: T = 2π√(L/g), where g is the acceleration due to gravity.

Therefore, we arrive at the equation: T = 2π√(L/g), which is the formula for the period of a simple pendulum.

Now that we have derived the equation, if you have the length (L) of the pendulum, you can plug it into the equation along with the acceleration due to gravity (g ≈ 9.8 m/s²) to find the period (T) of the pendulum.