91Ó°ÊÓ

Spring oscillations describe the motion of a mass connected to a spring, which moves back and forth in a periodic manner. This movement is fundamentally governed by Hooke's Law, which connects the forces acting on the spring to its displacement. When a spring is compressed or stretched, it exerts a force to return to its natural length. The oscillatory motion observed is a result of this restorative force, leading to simple harmonic motion.

A key aspect of spring oscillations is the frequency of the oscillation. This is determined by the spring constant, denoted as \( k \), and the mass of the object, \( m \). The general differential equation for describing the motion is \( m \frac{d^2s}{dt^2} + ks = 0 \), where \( s(t) \) indicates the displacement over time. This motion equation simplifies to a standard form, revealing the nature of the oscillations.

When a specific frequency is involved, denoted as \( \omega = \sqrt{\frac{k}{m}} \), it dictates how fast the oscillations occur. In our exercise, the angular frequency is \( \omega = \sqrt{2500} = 50 \), thus impacting the periodic nature of the movement.

Initial conditions play a crucial role in solving differential equations, especially in oscillatory systems like a mass-spring system. They help tailor the general solution of the differential equation to fit the particular scenario being studied.

For our spring oscillation problem, we assume that the mass starts at a specific position and with a known velocity. These values are essential for determining the constants in the motion equation solution.

• The initial position is given as \( s(0) = 5 \), resulting in the constant \( A = 5 \).
• The initial velocity is \( s'(0) = -10 \), which, from the derivative of the general solution, calculates \( B \).

These initial data points allow us to find the particular solution - the one that fits this unique starting situation: \( s(t) = 5 \cos(50t) - 0.2 \sin(50t) \).

Such information essentially "anchors" the solution, making it specific to the oscillatory behavior from that point onward.

Hooke's Law is foundational in understanding spring oscillations. It describes the force required to compress or extend a spring by a distance \( x \) from its equilibrium position. Expressed mathematically, Hooke's Law states \( F = -kx \), where \( F \) is the force, \( k \) is the spring constant, and \( x \) is the displacement.

This force acts in a direction opposite to the displacement, which is why the negative sign is present. It indicates that the force aims to bring the spring back to its original position.

• The spring constant \( k \) signifies the stiffness of the spring; a higher \( k \) value represents a stiffer spring.
• Hooke's law is linear; a doubling of displacement results in doubling of the force required.

In dynamic systems like the one in this spring problem, Hooke's Law helps establish the basis for the differential equation governing motion, \( m \frac{d^2s}{dt^2} + ks = 0 \). This equation then models how the spring system responds to displacements over time, pivotal for analyzing and predicting the system's behavior.