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.

Verified step by step guidance

1. Identify the relationship given: The period \(T\) varies directly as the square root of the length \(L\) and inversely as the square root of the acceleration due to gravity \(g\). This can be written as the formula: \[T = k \frac{\sqrt{L}}{\sqrt{g}}\] where \(k\) is the constant of proportionality.
2. Use the given values to find the constant \(k\). Substitute \(T = 6\pi\), \(L = 289\), and \(g = 980\) into the formula: \[6\pi = k \frac{\sqrt{289}}{\sqrt{980}}\]
3. Simplify the square roots in the equation: \[6\pi = k \frac{17}{\sqrt{980}}\] Then solve for \(k\) by multiplying both sides by \(\frac{\sqrt{980}}{17}\):
4. Now that you have \(k\), use it to find the period \(T\) when \(L = 121\) and \(g = 980\). Substitute these values into the original formula: \[T = k \frac{\sqrt{121}}{\sqrt{980}}\]
5. Simplify the square roots and substitute the value of \(k\) found earlier to express \(T\). This will give you the period for the new length and gravity values.

Key Concepts

Direct and Inverse Variation

Direct variation means one quantity increases as another increases, while inverse variation means one quantity decreases as another increases. In this problem, the period varies directly with the square root of the length and inversely with the square root of gravity, combining both types of variation in one formula.

Square Root Function

The square root function involves taking the root of a number, which affects how variables relate to each other. Here, the period depends on the square roots of length and gravity, meaning changes in these values affect the period non-linearly, requiring careful manipulation of roots in calculations.

Using Given Conditions to Find Constants

To solve variation problems, you use given values to find the constant of proportionality. By substituting known period, length, and gravity values into the variation formula, you determine the constant, which then allows calculation of the period for new values of length and gravity.