Final answer:
To express k in terms of T, h, and g for a compound pendulum, rearrange the formula T = 2π√(h² + k²/gh), isolate k, and solve. The final expression for k is √[(T²gh)/(4π²) - h²].
Step-by-step explanation:
The equation for the period of a compound pendulum is given by T = 2π√(h² + k²/gh). We want to express k in terms of T, h, and g. By manipulating the formula, we can isolate k to find the relation.
Steps:
Square both sides of the equation to get rid of the square root: T² = 4π²(h² + k²)/gh.
Multiply both sides by gh to clear the denominator: T²gh = 4π²(h² + k²).
Divide both sides by 4π² to simplify: (T²gh)/(4π²) = h² + k².
Subtract h² from both sides to isolate k²: (T²gh)/(4π²) - h² = k².
Finally, take the square root of both sides to solve for k: k = √[(T²gh)/(4π²) - h²].
k is now expressed in terms of T, h, and g, showing the relationship between the rod's radius of gyration, the period, and the gravitation.