The 15 g head of a bobble-head doll oscillates in SHM at a...

The 15 g head of a bobble-head doll oscillates in $\mathrm{SHM}$ at a frequency of $4.0 \mathrm{Hz}$ a. What is the spring constant of the spring on which the head is mounted? b. Suppose the head is pushed $2.0 \mathrm{cm}$ against the spring, then released. What is the head's maximum speed as it oscillates? c. The amplitude of the head's oscillations decreases to $0.5 \mathrm{cm}$ in $4.0 \mathrm{s}$. What is the head's damping constant?

The 15 g head of a bobble-head doll oscillates in $\mathrm{SHM}$ at a frequency of $4.0 \mathrm{Hz}$
a. What is the spring constant of the spring on which the head is mounted?
b. Suppose the head is pushed $2.0 \mathrm{cm}$ against the spring, then released. What is the head's maximum speed as it oscillates?
c. The amplitude of the head's oscillations decreases to $0.5 \mathrm{cm}$ in $4.0 \mathrm{s}$. What is the head's damping constant?

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Key Concepts

Simple Harmonic Motion (SHM)

This is a type of periodic motion where the restoring force is proportional to the negative of the displacement from an equilibrium position. It is characterized by constant oscillation frequency, amplitude, and period, governed by differential equations that lead to sinusoidal solutions.

Mass-Spring System

A mass-spring system is a classical mechanical model used to illustrate SHM where a mass is connected to a spring obeying Hooke’s law. The system’s dynamics, such as the oscillation frequency, are determined by the mass and the spring constant, with the angular frequency defined by ? = ?(k/m).

Angular Frequency and Maximum Speed

Angular frequency measures how fast the system oscillates and relates to the physical frequency of the motion. In SHM, the maximum speed is achieved when the mass passes through the equilibrium position and is found by multiplying the angular frequency by the motion’s amplitude.

Damped Oscillations

Damped oscillations occur when non-conservative forces, such as friction or air resistance, cause the amplitude of oscillation to decay over time. The damping constant quantifies this rate of energy loss and is used in the exponential decay formula that describes how quickly the oscillations diminish.

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