To determine the spring constant $k$, we use the relationship between the angular frequency $\omega$ and the spring constant for a mass-spring system:

$\omega = \sqrt{\frac{k}{m}}$

where:

• $\omega$ is the angular frequency (given as $42.5 \, \text{rad/s}$),
• $m$ is the mass of the particle ($0.110 \, \text{kg}$).

Rearranging for $k$, we have:

$k = m \omega^2$

Substituting the given values:

$k = 0.110 \, \text{kg} \cdot (42.5 \, \text{rad/s})^2$

$k = 0.110 \cdot 1806.25$

$k = 198.69 \, \text{N/m}$

Final Answer:

The spring constant $k$ is approximately 198.7 N/m.

Would you like further explanations or details about this calculation?

Here are 5 related questions:

1. How is the angular frequency related to the spring constant in simple harmonic motion?
2. What is the total mechanical energy of the system using the given data?
3. How does the displacement equation describe the motion of the particle?
4. What is the maximum velocity and acceleration