Simple Pendulum — Changing the Period If the period T of a simple pendulum is to be doubled, by what factor should its length l be changed (small-angle assumption)?

Introduction / Context:
The period of a simple pendulum depends on its length and the local acceleration due to gravity. Understanding how T changes with l is essential for timing devices and lab experiments using pendulums.

Given Data / Assumptions:

• Small-angle approximation holds so that T = 2π * sqrt(l / g).
• Gravity g is constant.
• We desire T_new = 2 * T_old.

Concept / Approach:
Since T ∝ sqrt(l), scaling the period by a factor scales the length by the square of that factor. Therefore doubling T requires quadrupling l.

Step-by-Step Solution:

Verification / Alternative check:
If l is quadrupled, T becomes 2π * sqrt(4 l_old / g) = 2 * 2π * sqrt(l_old / g), confirming the doubling.

Why Other Options Are Wrong:
Halving or doubling l changes T by factors of 1/√2 or √2 respectively, not 2.

Common Pitfalls:
Forgetting that T depends on sqrt(l), not linearly; overlooking that mass does not affect T for a simple pendulum.

Final Answer:
quadrupled.