A simple pendulum is vibrating with an angu amplitude of 90^∘ as shown in the figure. For what v

step 1 The question is simple pendulum is vibrating with an angular amplitude of 90 degrees shown in th

A simple pendulum is vibrating with an angu amplitude of...

A simple pendulum is vibrating with an angu amplitude of $90^{\circ}$ as shown in the figure. For what va of $\alpha$, the acceleration is directed horizontally?
(a) 0
(b) $90^{\circ}$
(c) $\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
(d) $\sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Key Concepts

Components of Acceleration in Pendulum Motion

In pendulum dynamics, the acceleration of the bob can be decomposed into two components: the tangential acceleration, which is responsible for speeding up or slowing down the bob along the circular arc, and the centripetal (or radial) acceleration, which points towards the pivot and maintains the circular path. Understanding the interplay between these components is essential for analyzing the direction and magnitude of the total acceleration at any point in the motion.

Force Decomposition and Direction of Acceleration

The gravitational force acting on the pendulum bob can be resolved into components along the tangential and radial directions. The tangential component of gravity gives rise to the restoring acceleration, while its projection in various directions determines when the net acceleration might align horizontally. Analyzing these components allows us to determine specific angular positions where the acceleration vector points in the horizontal direction.

Simple Pendulum

A simple pendulum is a system comprised of a point mass suspended from a fixed pivot by a lightweight, inextensible string. It oscillates under the influence of gravity, and its motion is typically analyzed by resolving forces into components and applying Newton’s laws. This system serves as an idealized model to understand periodic motion and small oscillations.

Angular Amplitude

Angular amplitude refers to the maximum angular displacement from the vertical position that the pendulum bob reaches during its swing. It defines the limits of the pendulum’s motion and is critical in determining the behavior of the restoring force, which is proportional to the sine of the angular displacement.

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