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The spring constant, often denoted by the symbol \( k \), is a crucial parameter in a mass-spring system. It measures the stiffness of the spring and is expressed in newtons per meter (N/m).
The higher the spring constant, the stiffer the spring is, meaning it requires more force to stretch or compress by a certain distance.
In the exercise, the spring constant is given as \( 112 \, \text{N/m} \), meaning the spring is reasonably stiff. This stiffness affects both the frequency of oscillation and the potential energy stored in the system when the spring is compressed or stretched.
Understanding the spring constant helps in determining how the system will behave under various forces, including how quickly it will oscillate and the maximum amplitude of these oscillations.

Frequency of oscillation refers to how many cycles an oscillating system completes in one second.
In simple harmonic motion, the frequency \( f \) can be determined by the formula:
\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass.
For the given problem, substituting \( k = 112 \, \text{N/m} \) and \( m = 0.400 \, \text{kg} \), we calculate:
\[ f \approx 2.66 \, \text{Hz} \]
This means the block, once pushed, will complete about 2.66 cycles per second.
Understanding frequency is vital in predicting the behavior of a mass-spring system, especially when designed to work acoustically or mechanically.

Amplitude in the context of simple harmonic motion refers to the maximum extent of movement of the oscillating object from its equilibrium position.
The amplitude is a crucial factor because it indicates the energy level of the system and can affect whether an object loses contact with the spring in vertical motions.
According to the problem, the amplitude at which the block loses contact is where the spring force can no longer counteract gravity. The condition for losing contact occurs when:

• The spring force and gravitational force balance (i.e., \( ma = mg \)).
• This leads to the calculation: \( A = \frac{g}{\omega^2} \)
• With calculated amplitude \( A \approx 0.035 \)

Thus, at approximately \( 0.035 \text{m} \), the block will start to lose contact with the spring as it oscillates.

The mass-spring system is a classic example of simple harmonic oscillation, where an object of mass \( m \) is attached to a spring with stiffness \( k \).
In this system, the block and spring work together to create repetitive motion back and forth around an equilibrium position.
The two opposing forces involved are the restoring force of the spring, which tries to return the mass to the equilibrium position, and the inertial force of the mass, which drives it away.
Key characteristics of such systems include:

• The natural frequency of oscillation, determined by both the mass and spring constant.
• The potential for energy transfer between kinetic and potential energy during oscillation.
• The ability to apply these principles to many real-world systems, such as car suspensions and measuring scales.

Understanding these dynamics allows for predictive modeling and analysis of many mechanical systems using just a mass and a spring.