91Ó°ÊÓ

The spring constant, denoted as \(\text{k}\), represents the stiffness of the spring in a harmonic oscillator system. It is measured in Newtons per meter (\(\text{N/m}\)). In the given problem, the spring constant is provided as \(k = 400 \text{N/m}\). This value tells us how much force is required to stretch or compress the spring by a unit distance. For instance, if a force of 400 Newtons stretches the spring by one meter, this relationship is directly proportional, and it also affects the system's oscillation properties. This value plays a crucial role in determining other parameters such as the angular frequency and the system's total energy.

Angular frequency, represented by \(\omega\), is a vital concept in understanding oscillatory motion. It is defined as the rate at which an object moves through its cycle of motion in radians per second (\(\text{rad/s})\). For a spring-mass system, the angular frequency is given by the formula: \(\text{\omega}=\sqrt{\frac{k}{m}}\). This formula highlights the dependence of angular frequency on both the spring constant (\(\text{k})\) and the mass of the block (\(m)\). Once we calculate \(\omega\), we can also determine the frequency of oscillation \(\mathbf{f}\) using the equation: \[f = \frac{\omega}{2\pi}\]. For example, if our spring constant is 400 N/m and we know the mass of the block, we can substitute these values into the formula to find \(\omega\). This value is crucial for solving various aspects of the problem including the system's energy and period of oscillation.

Newton's second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration, expressed as \(\mathbf{F} = m a\). In our oscillatory system, this law helps us relate the spring force to the block's mass and acceleration. The spring force can also be described by Hooke's Law, which states that \(\mathbf{F} = -k x\), where \(\text{k}\) is the spring constant and \(\text{x}\) is the displacement from equilibrium. By combining these two laws, we can derive the formula \(\text{a} = \frac{-kx}{m}\). Rearranging this equation allows us to solve for the mass of the block: \(\text{m} = \frac{-kx}{a}\). For instance, with the given values \(k=400\text{N/m}\text{, x=0.100 m, and a=-123 m/s^2}\), we substitute these into the equation to find the mass.

The amplitude of oscillation, \(\mathbf{A}\), is the maximum displacement from the equilibrium position in an oscillatory motion. It is an essential parameter that describes the extent of motion in a harmonic oscillator. To calculate the amplitude, we use the formula: \(A = \sqrt{x^2 + \left(\frac{v^2}{\omega^2}\right)}\). Here, \(\text{x}\) represents the displacement at any given moment, \(\text{v}\) is the velocity, and \(\omega\) is the angular frequency obtained from previous calculations. By plugging in the given values \(x = 0.100 m\), \(v=-13.6 \text{m/s},\text{ and}\text{, and \(\omega\)}\) determined earlier, we get the amplitude. Understanding amplitude is crucial as it tells us the 'energy' in the system and affects the kinetic and potential energy calculations.