The bob of a simple pendulum of length L is released at time...

The bob of a simple pendulum of length $L$ is released at time $t=0$ from a position of small angular displacement. Its linear displacement at time $t$ is given by (b) $x=a \cos 2 \pi \sqrt{\frac{g}{L}} \times t$ $X=a \sin 2 \pi \sqrt{\frac{L}{g}} \times t$ (c) $X=a \sin \sqrt{\frac{g}{L}} \times t$ (d) $X=a \cos \sqrt{\frac{g}{L}} \times t$

The bob of a simple pendulum of length $L$ is released at time $t=0$ from a position of small angular displacement. Its linear displacement at time $t$ is given by
(b) $x=a \cos 2 \pi \sqrt{\frac{g}{L}} \times t$
$X=a \sin 2 \pi \sqrt{\frac{L}{g}} \times t$
(c) $X=a \sin \sqrt{\frac{g}{L}} \times t$
(d) $X=a \cos \sqrt{\frac{g}{L}} \times t$

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass suspended from a fixed point by a string or rod, where the motion is confined to a plane. Its analysis involves examining the restoring force due to gravity acting on the pendulum bob, resulting in oscillatory motion, particularly when the displacement is small.

Small Angle Approximation

For small angular displacements, the sine of the angle is approximately equal to the angle in radians. This approximation simplifies the nonlinear equations of motion of the pendulum into a linear form, making it possible to treat the motion as simple harmonic.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes oscillatory systems where the restoring force is directly proportional to the displacement and acts in the opposite direction. Under the small angle approximation, a simple pendulum exhibits SHM, allowing its motion to be expressed using sine or cosine functions.

Angular Frequency

The angular frequency in the context of a simple pendulum is the rate of oscillation and is given by the square root of the ratio of gravitational acceleration to pendulum length (?(g/L)). This parameter determines the period and the dynamics of the oscillatory motion.

Phase in Oscillatory Motion

The use of sine or cosine functions in expressing the displacement reflects the phase of the oscillation, which is determined by the initial conditions of the motion. The choice of function and its argument, including the angular frequency multiplied by time, describes how the system evolves over time.

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