What determines the natural frequency of an oscillator?

The natural frequency of an oscillator is determined by the system's mass and the stiffness of the spring.

The natural frequency, often referred to as the resonant frequency, is a key characteristic of oscillatory systems, such as a mass-spring system. It is the frequency at which the system naturally tends to oscillate in the absence of any driving or damping force. The natural frequency of such a system is primarily determined by two factors: the mass of the object and the stiffness of the spring.

The mass of the object, often denoted as 'm', plays a crucial role. The heavier the mass, the slower the system tends to oscillate, resulting in a lower natural frequency. This is because a larger mass requires more force to change its motion, according to Newton's second law of motion. Therefore, for a given force, a larger mass will accelerate less and hence oscillate slower.

On the other hand, the stiffness of the spring, usually represented by the spring constant 'k', also influences the natural frequency. A stiffer spring (one with a larger spring constant) will cause the system to oscillate faster, resulting in a higher natural frequency. This is because a stiffer spring exerts a larger restoring force for a given displacement, causing the mass to accelerate more and hence oscillate faster.

The relationship between the natural frequency, the mass, and the spring constant is given by the formula: f = 1/(2π) * √(k/m), where 'f' is the natural frequency, 'k' is the spring constant, and 'm' is the mass. This formula shows that the natural frequency is directly proportional to the square root of the stiffness and inversely proportional to the square root of the mass.

In summary, the natural frequency of an oscillator is a fundamental property of the system that is determined by the mass of the object and the stiffness of the spring. A larger mass will lower the natural frequency, while a stiffer spring will increase it.