The 20 -cm-long wrench swings on its hook with a period of...

The 20 -cm-long wrench swings on its hook with a period of $0.90 \mathrm{s}$. When the wrench hangs from a spring of spring constant $360 \mathrm{N} / \mathrm{m},$ it stretches the spring $3.0 \mathrm{cm} .$ What is the wrench's moment of inertia about the hook?
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Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) refers to periodic motion where the restoring force is proportional to the displacement. Both mass-spring systems and pendular systems can exhibit SHM under small displacement conditions, and this principle is used to derive relationships (such as the period formula for a physical pendulum) that connect the system’s inertia and its oscillatory behavior.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to angular acceleration about a given axis and is determined by the mass distribution relative to that axis. In problems involving oscillatory motion of a pendulum, the moment of inertia plays a crucial role in determining the period of oscillation and is calculated using the dynamics of rotational motion.

Physical Pendulum

A physical pendulum is a rigid body that oscillates about a pivot point under the influence of gravity. Its motion is described by the formula T = 2??(I/(mgd)), where I is the moment of inertia about the pivot, m is the mass, g is the acceleration due to gravity, and d is the distance from the pivot to the center of mass. Understanding this concept is key to linking the observed period of oscillation to the rotational inertia of the body.

Hooke’s Law

Hooke’s law states that the force exerted by a spring is directly proportional to its displacement from equilibrium (F = k·?x). This law is fundamental in determining the mass of an object when its weight results in a measurable stretch in a spring, as the gravitational force (mg) is balanced by the spring’s restoring force.

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