The equation of motion for a spring-mass system excited by a harmonic force is

                                       Mẍ + Kx = F cos(ωt),

where M is the mass, K is the spring stiffness, F is the force amplitude and ω is the angular frequency of excitation. Resonance occurs when ω is equal to

The correct answer is $\sqrt{\frac{K}{M}}$.

Step-by-Step Explanation:

1. Identify the System's Equation of Motion:
The equation of motion for the spring-mass system is given by:

$M\stackrel{··}{x}+Kx=F\cos \left(\omega t\right)$

where:
• $M$ represents the mass of the system.
• $K$ represents the spring stiffness.
• $F$ is the excitation force amplitude.
• $\omega$ is the excitation angular frequency.
• $\stackrel{··}{x}$ represents the acceleration ($\frac{d^{2}x}{d{t}^2}$).

2. Determine the Natural Angular Frequency:
For a free, undamped spring-mass system, the natural angular frequency (${\omega }_n$) is derived from the homogeneous part of the differential equation:

$M\stackrel{··}{x}+Kx=0$

Solving this gives the natural angular frequency as:

${\omega }_n=\sqrt{\frac{K}{M}}$

3. Understand the Condition for Resonance:
Resonance is a phenomenon that occurs when the frequency of the externally applied harmonic force matches the natural frequency of the vibrating system. In this state, the system oscillates with maximum amplitude.
Mathematically, resonance occurs when the excitation frequency $\omega$ is equal to the natural angular frequency ${\omega }_n$:

$\omega ={\omega }_n=\sqrt{\frac{K}{M}}$