A 5.0 kg block hangs from a spring with spring constant 2000...

A $5.0 \mathrm{kg}$ block hangs from a spring with spring constant $2000 \mathrm{N} / \mathrm{m} .$ The block is pulled down $5.0 \mathrm{cm}$ from the equilibrium position and given an initial velocity of $1.0 \mathrm{m} / \mathrm{s}$ back toward equilibrium. What are the (a) frequency, (b) amplitude, and (c) total mechanical energy of the motion?

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. It is characterized by sinusoidal oscillations and is governed by differential equations that yield predictable behavior such as constant frequency and amplitude.

Mass-Spring System

A mass-spring system is a classic example of SHM in which a mass attached to a spring oscillates when displaced from its equilibrium position. The system’s behavior is determined by the mass and the spring constant, where the latter defines the stiffness of the spring and influences the frequency of oscillation.

Angular Frequency and Frequency

Angular frequency (?) is a measure of how quickly an object oscillates in radians per second and is related to the mass and spring constant in a mass-spring system by the relationship ? = ?(k/m). The ordinary frequency (f) in cycles per second is then obtained by dividing the angular frequency by 2?.

Amplitude Determination in SHM

Amplitude in simple harmonic motion represents the maximum displacement from the equilibrium position. It is determined by both the initial displacement and the initial velocity of the oscillating object. In physics problems, initial conditions are used to compute the amplitude, ensuring that all energy contributions are accounted for.

Mechanical Energy in Oscillatory Motion

The total mechanical energy in a mass-spring system undergoing SHM is conserved in the absence of non-conservative forces like friction. This energy is the sum of kinetic energy and elastic potential energy, given by expressions involving the mass, velocity, spring constant, and displacement, and remains constant throughout the motion.

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