Final answer:
The formula T = 2π/√(r²/rg) tanx, representing the period of oscillation in terms of angle, can be derived using simple harmonic motion principles and by analyzing the projection of uniform circular motion. The period T is obtained by dividing the circumference of the circle by the tangential velocity. This derivation highlights the connection between circular motion and simple harmonic motion.
Step-by-step explanation:
The derivation of the formula T = 2π/√(r²/rg) tanx, where T represents the period of oscillation and X denotes the angle, can be done using Simple harmonic motion principles. To derive this formula, we consider a projection of uniform circular motion which undergoes simple harmonic oscillation. By analyzing the projection of uniform circular motion, we can obtain all the characteristics of simple harmonic motion.
The period T can be found by considering the time it takes for a point on the edge of the circle to complete one revolution. This time is equal to the circumference of the circle (2πr) divided by the tangential velocity (Umax = rg). Therefore, the period T is given by T = 2π/√(r²/rg) tanx.
This derivation is based on the relationship between circular motion and simple harmonic motion, allowing us to determine the period of oscillation based on the properties of the circular motion.