91Ó°ÊÓ

The spring constant, denoted as \( k \), is a measure of a spring's stiffness. It tells us how much force is needed to compress or extend the spring by a certain amount. The higher the spring constant, the stiffer the spring. In the context of simple harmonic motion (SHM), understanding the spring constant is crucial as it influences the oscillation behavior.
To find the spring constant from given frequency, the relationship used is:

• \( f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \)

Here, \( f \) is the frequency, and \( m \) is the mass of the oscillating object. By rearranging the formula, we can solve for \( k \):

• \( k = (2\pi f)^2 m \)

In the sleigh example, substituting in the values \( f = 0.225 \, \text{Hz} \) and \( m = 250 \, \text{kg} \) gives a spring constant of approximately \( 50.21 \, \text{N/m} \). This value indicates how strong the spring's pulling force is within this system.

Angular frequency, denoted by \( \omega \), describes how quickly an object completes a cycle of motion in SHM, measured in radians per second. It provides a different perspective than regular frequency, focusing more on the rotational aspects of motion, even in linear systems like the sleigh.
To compute angular frequency, the formula involved is:

• \( \omega = 2\pi f \)

Using the given frequency \( f = 0.225 \, \text{Hz} \), we calculate \( \omega \) as:

• \( \omega = 2\pi \times 0.225 \approx 1.414 \, \text{rad/s} \)

With angular frequency, we can then determine other motion characteristics, like maximum speed and acceleration during SHM. Angular frequency is essential for linking the periodic nature of oscillatory motion with its measurement in radians.

Maximum acceleration in the context of SHM indicates the largest magnitude of acceleration that an object can experience during its back-and-forth motion. It occurs at the extremes of motion when the object changes direction. In this scenario, maximum acceleration is synonymous with the force exerted being highest, due to its dependence on both the angular frequency and amplitude.
The formula to determine maximum acceleration \( a_{max} \) is:

• \( a_{max} = \omega^2 A \)

Here, \( \omega \) is the angular frequency, and \( A \) is the amplitude of motion. Using \( \omega = 1.414 \, \text{rad/s} \) and \( A = 0.950 \, \text{m} \), the maximum acceleration is:

• \( a_{max} = (1.414)^2 \times 0.950 \approx 1.895 \, \text{m/s}^2 \)

Understanding maximum acceleration is crucial in designing systems to ensure they can handle such forces without failure or loss of performance.