the angle made by string of simple pendulum with the vertical depends on time as θ=π/90 sin^-1[(π s^-1) t]. find length pendulum if g=π^2

To find the length of the pendulum given the angle θ as a function of time, we can start by analyzing the equation provided: θ = (π/90) sin^(-1)[(π s^(-1)) t]. This equation describes how the angle changes over time, and we can relate it to the properties of a simple pendulum.

Understanding the Pendulum Motion

A simple pendulum consists of a mass (the bob) attached to a string of length L, swinging back and forth under the influence of gravity. The motion of a pendulum can be approximated as simple harmonic motion when the angles are small. The period T of a simple pendulum is given by the formula:

T = 2π√(L/g)

Where g is the acceleration due to gravity. In this case, we are given that g = π².

Relating the Angle to Time

The angle θ can be expressed in terms of the maximum angle (amplitude) and the time variable. The equation you provided suggests that the angle varies sinusoidally with time, which is typical for pendulum motion. The maximum angle (amplitude) can be derived from the equation:

θ_max = π/90

From the equation θ = (π/90) sin^(-1)[(π s^(-1)) t], we can see that the term (π s^(-1)) t represents the argument of the sine function, which indicates how the angle changes over time.

Finding the Length of the Pendulum

To find the length L of the pendulum, we need to relate the period T to the angular frequency ω. The angular frequency ω can be derived from the equation of motion:

ω = π s^(-1)

The relationship between angular frequency and the period is given by:

ω = 2π/T

From this, we can rearrange to find T:

T = 2π/ω = 2π/(π s^(-1)) = 2/s

Now, substituting T back into the period formula for a pendulum:

2/s = 2π√(L/(π²))

To simplify, we can square both sides:

(2/s)² = 4/π² = 4L/π²

Now, solving for L gives:

L = (4/π²) * (π²/4) = 1

Final Result

Thus, the length of the pendulum is L = 1 meter. This result shows how the pendulum's length can be derived from the relationship between the angle, time, and the acceleration due to gravity. Understanding these relationships is crucial for analyzing pendulum motion and similar systems in physics.