PHYSICS - PENDULUM A simple pendulum is formed by hanging a...

PHYSICS - PENDULUM A simple pendulum is formed by hanging a bob of mass $M$ on a string of length $L$ from a fixed support (see the figure). The time it takes the bob to swing from right to left and back again is called the period $T$ and is given by $$ T=2 \pi \sqrt{\frac{L}{g}} $$ where $g$ is the gravitational constant. Show that $T$ can be written in the form $$ T=\frac{2 \pi \sqrt{g L}}{g} $$

PHYSICS - PENDULUM A simple pendulum is formed by hanging a bob of mass $M$ on a string of length $L$ from a fixed support (see the figure). The time it takes the bob to swing from right to left and back again is called the period $T$ and is given by
$$
T=2 \pi \sqrt{\frac{L}{g}}
$$
where $g$ is the gravitational constant. Show that $T$ can be written in the form
$$
T=\frac{2 \pi \sqrt{g L}}{g}
$$

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Key Concepts

Simple Pendulum

A simple pendulum consists of a mass (or bob) suspended from a fixed point by a string or rod that swings freely under the influence of gravity. Its motion, when the oscillations are small, approximates simple harmonic motion, making it an essential model for studying periodic systems in physics.

Period of Oscillation

The period of a simple pendulum is the time it takes to complete one full cycle of motion, defined by the formula T = 2??(L/g) where L is the length of the pendulum and g is the acceleration due to gravity. This relationship shows how the period depends solely on the length and the gravitational constant, independent of the mass of the bob.

Algebraic Manipulation using Properties of Square Roots

The process of rewriting the period formula from T = 2??(L/g) to the equivalent form T = (2??(gL))/g relies on properties of square roots and algebraic manipulation. By recognizing that ?(L/g) can be re-expressed as ?L/?g and then further manipulated, one can demonstrate the algebraic equivalency of the two forms. This technique is fundamental in mathematical problem solving and helps verify the consistency of physical formulas.

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