Period of a Pendulum The period of a pendulum varies directly as the square root of the length of the pendulum and inversely as the square root of the acceleration due 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 equation: \(T = k \frac{\sqrt{L}}{\sqrt{g}}\) 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 equation: \$6\pi = k \frac{\sqrt{289}}{\sqrt{980}}$
3. Solve the equation for \(k\) by isolating it on one side: \(k = 6\pi \times \frac{\sqrt{980}}{\sqrt{289}}\)
4. Now, use the value of \(k\) found in the previous step 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. Substitute the expression for \(k\) from step 3 into the equation in step 4 to express \(T\) entirely in terms of known quantities, then simplify to find the period.

Key Concepts

Direct and Inverse Variation

Direct variation means one quantity increases as another increases, often expressed as y = kx. Inverse variation means one quantity increases as another decreases, expressed as y = k/x. In this problem, the period varies directly with the square root of length and inversely with the square root of gravity, combining both types of variation.

Square Root Function

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

Solving for Constants in Variation Problems

To solve variation problems, first express the relationship with a constant k, then use given values to find k. Once k is known, substitute new values to find the unknown quantity. This method allows solving for the period given different lengths and gravity values.