91Ó°ÊÓ

In an oscillating block-spring system, the spring constant (\(k\)) is a measure of the stiffness of the spring. It tells us how much force is needed to stretch or compress the spring by a certain distance. A stiffer spring has a higher spring constant.
The spring constant is crucial in determining the dynamics of the oscillating system. It's calculated using the formula for mechanical energy in a spring-mass system, which is \(E = \frac{1}{2} k A^2\). Here, \(E\) is the mechanical energy and \(A\) is the amplitude.
In the given problem, we solved for \(k\) by rearranging the equation to get:- \(E = \frac{1}{2} k A^2\)By substituting the known values:- \(1.00 = \frac{1}{2} k (0.10)^2\)We multiplied both sides by 2 and divided by \(0.01\) to find:- \(k = 200\,\text{N/m}\)This result shows the spring constant, telling us how robust the spring is against deformation.

Mechanical energy in a block-spring system refers to the total energy due to both kinetic and potential energy.
This energy is constant in an ideal system with no external forces or energy losses. The potential energy is stored in the spring when it is compressed or stretched, and the kinetic energy is possessed by the block due to its motion.
The formula for mechanical energy when considering the amplitude \(A\) of the oscillation is- \(E = \frac{1}{2} k A^2\)This formula helps clarify how energy relates to the displacement of the spring.
It also shows that the mechanical energy depends on both the spring constant and amplitude. Since energy is conserved in the system, you can use it to solve for other properties if some values are known, just like finding the spring constant in this problem.

Amplitude is the maximum extent of a vibration or oscillation, measured from the center position. In the context of the block-spring system, it represents the furthest point the block moves from the equilibrium position.
It is a key variable in calculating the mechanical energy, given by- \(E = \frac{1}{2} k A^2\)Here, the amplitude \(A\) (given as 10.0 cm or 0.10 m) directly influences how much energy the system holds. A larger amplitude means more energy is stored.
The amplitude doesn't change with time in an ideal system, as energy is conserved. Instead, it represents the scale of oscillation. Understanding amplitude helps visualize the system's overall motion and energetic properties.

Frequency of oscillation \(f\) refers to how many complete cycles of motion occur in a given time period, usually a second. In the block-spring system, frequency describes how quickly the system oscillates back and forth.
Frequency is important because it tells you how fast the system vibrates. The formula to calculate frequency in a block-spring system is:- \(f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}\)Where:- \(k\) is the spring constant- \(m\) is the mass of the block
In this problem, we used the calculated spring constant and mass to find the frequency:- \(f \approx 1.91\,\text{Hz}\)This means the block completes about 1.91 full oscillations per second. Frequency gives insights into the natural oscillation characteristics of the system, impacted by both the spring's stiffness and the block's mass.