Define natural frequency in oscillations.

The natural frequency in oscillations is the frequency at which a system oscillates without any external force.

When a system is disturbed from its equilibrium position and left to oscillate freely, it does so at a certain frequency known as its natural frequency. This frequency is determined by the system's physical properties such as its mass, stiffness, and damping. The natural frequency can be calculated using the formula:

ω = √(k/m)

where ω is the natural frequency in radians per second, k is the spring constant in newtons per meter, and m is the mass in kilograms.

For example, consider a mass-spring system with a mass of 0.5 kg and a spring constant of 10 N/m. The natural frequency of this system can be calculated as follows:

ω = √(10/0.5) = √20 = 4.47 rad/s

This means that the system will oscillate at a frequency of 4.47 cycles per second (or hertz) when disturbed from its equilibrium position and left to oscillate freely.

The natural frequency is an important concept in many areas of physics and engineering, as it determines the behavior of systems such as bridges, buildings, and electronic circuits. Understanding the natural frequency of a system can help engineers design structures and devices that are stable and efficient.