A 1.0 kg block is attached to a spring with spring constant...

A $1.0 \mathrm{kg}$ block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of $40 \mathrm{cm} / \mathrm{s}$ What are a. The amplitude of the subsequent oscillations? b. The block's speed at the point where $x=\frac{1}{2} A ?$

A $1.0 \mathrm{kg}$ block is attached to a spring with spring constant 16 N/m. While the block is sitting at rest, a student hits it with a hammer and almost instantaneously gives it a speed of $40 \mathrm{cm} / \mathrm{s}$ What are
a. The amplitude of the subsequent oscillations?
b. The block's speed at the point where $x=\frac{1}{2} A ?$

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Key Concepts

Hooke's Law and Spring Potential Energy

This concept states that the restoring force of a spring is proportional to its displacement from its equilibrium position, expressed as F = -kx, where k is the spring constant. The potential energy stored in the spring at a displacement x is given by (1/2)kx². Understanding this principle is essential when analyzing energy exchanges in a mass-spring system.

Conservation of Mechanical Energy

In the absence of non-conservative forces like friction, the total mechanical energy in the system remains constant. This means that the initial kinetic energy imparted to the block can be completely converted into the spring's potential energy at the amplitude, and vice versa. This principle allows one to determine the amplitude of oscillations and relate the speed at various displacements through energy conversion equations.

Simple Harmonic Motion Kinematics

Simple harmonic motion (SHM) describes the oscillatory movement of a mass-spring system, where the mass alternately converts kinetic energy into potential energy. The kinematics of SHM, including expressions for displacement and velocity (such as v = ??(A² - x²), where ? = ?(k/m) is the angular frequency), help in determining the system's behavior at any given point in its oscillation, including calculating speeds at specified displacements.

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