Solve each problem. Period of a PendulumThe period of a pendulum varies directly as the square rootof the length of the pendulum and inversely as the square root of the accelerationdue to gravity. Find the period when the length is 121 cm and the acceleration due to gravity is 980 cm per second squared, if the period is 6π seconds when the length is 289 cm and the acceleration due to gravity is 980 cm per second squared.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another. In the context of the pendulum's period, it indicates that as the length of the pendulum increases, the period also increases proportionally. This relationship can be expressed mathematically as P = k√L, where P is the period, L is the length, and k is a constant.

Inverse Variation

Inverse variation occurs when one variable increases while another decreases, maintaining a constant product. For the pendulum's period, this means that as the acceleration due to gravity increases, the period decreases. This relationship can be expressed as P = k/√g, where g is the acceleration due to gravity, and k is a constant.

Combining Variations

Combining direct and inverse variations allows us to create a comprehensive formula that accounts for both relationships. In this case, the period of the pendulum can be expressed as P = k√(L/g). This formula enables us to calculate the period for different lengths and gravitational accelerations by determining the constant k from known values.