A 0.25 kg body oscillates in SHM on a spring that, when extended 2.0 mm from its equilibrium p

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A 0.25 kg body oscillates in SHM on a spring that, when...

A $0.25 \mathrm{~kg}$ body oscillates in SHM on a spring that, when extended $2.0 \mathrm{~mm}$ from its equilibrium position, has an $8.0 \mathrm{~N}$ restoring force. What are (a) the angular frequency of oscillation, (b) the period of oscillation, and (c) the capacitance of an $L C$ circuit with the same period if $L$ is $5.0 \mathrm{H}$ ?

Key Concepts

Hooke's Law

Hooke's Law states that the restoring force exerted by a spring is directly proportional to its displacement from equilibrium, expressed as F = -kx. This principle allows for the determination of the spring constant (k) by dividing the applied force by the displacement.

Simple Harmonic Motion in a Spring-Mass System

Simple harmonic motion (SHM) describes the oscillatory motion of systems like a mass attached to a spring, where the acceleration is proportional to the negative of the displacement. In such systems, key characteristics such as angular frequency depend on the mass and spring constant, illustrating the predictability of the motion.

Angular Frequency and Period

Angular frequency (?) in SHM is related to the system’s parameters by the formula ? = ?(k/m), where k is the spring constant and m is the mass. The period (T) of oscillation, which is the time for one complete cycle of motion, is directly connected to ? by T = 2?/?, highlighting the inverse relationship between frequency and period.

LC Circuit Resonance

An LC circuit, comprising an inductor (L) and a capacitor (C), oscillates with a natural frequency given by ? = 1/?(LC). This relationship, analogous to the mass-spring system, allows one to find one component (for example, capacitance) when the period or frequency of the circuit and the other component (inductance) are known, emphasizing the fundamental similarities in oscillatory behavior across different physical systems.

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