A horizontal plank of mass m and length L is pivoted at one...

A horizontal plank of mass $m$ and length $L$ is pivoted at one end. The plank's other end is supported by a spring of force constant $k$ (Fig. PIS. $P 15.57 )$ . The plank is displaced by a small angle $\theta$ from its horizontal equilibrium position and released. Find the angular frequency with which the plank moves with simple harmonic motion.

Key Concepts

Moment of Inertia

The moment of inertia is a measure of how mass is distributed about the axis of rotation and determines the resistance of an object to changes in its rotational motion. For a plank or rod pivoted at one end, calculating the moment of inertia is essential because it directly influences the angular frequency of oscillations, linking the mass distribution with dynamic responses to applied torques.

Simple Harmonic Motion (SHM) in Rotational Systems

SHM describes motion where the restoring force (or torque in a rotational system) is directly proportional to the displacement from equilibrium and acts in the opposite direction. In the context of the plank and spring system, this principle is used to derive the angular frequency of oscillation by relating the effective spring constant (or torque per unit angle) to the moment of inertia of the plank.

Small Angle Approximation

When displacements are small, the sine of the angle can be approximated by the angle itself (in radians). This approximation simplifies the mathematical treatment of the problem by linearizing the system, allowing the restoring torque to be directly proportional to the angular displacement, which is a fundamental requirement for a simple harmonic oscillator.

Rotational Dynamics

This concept involves the study of how torque causes angular acceleration in a rigid body. For any rotating object, the net torque about an axis is equal to the moment of inertia times the angular acceleration. This relationship is analogous to Newton’s second law for linear motion and is central to analyzing the rotational motion of the plank, where the restoring torque provided by the spring sets the plank into simple harmonic motion.

Torque and Restoring Forces

Torque is the rotational equivalent of force and is generated when forces are applied at a distance from the pivot point. In this problem, the spring applies a force at the end of the plank which produces a torque about the pivot. The nature of the restoring torque—proportional to the displacement (in this case, the small angle)—is what leads to simple harmonic motion, enabling the determination of an angular frequency.

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