When the frequency of a force applied to a system matches the natural frequency of vibration of the system, it starts vibrating with a:

(I believe its larger amplitude)

Correct! When the frequency of a force applied to a system matches the natural frequency of vibration of the system, it can result in a phenomenon called resonance. In this case, the system will vibrate with a larger amplitude.

To understand why this happens, let's explain the concept of natural frequency and resonance.

Every object or system has a natural frequency of vibration, which is the frequency at which it naturally oscillates or vibrates when disturbed. When a force is applied to a system at its natural frequency, it causes the system to vibrate with the greatest amplitude possible.

To find the natural frequency of a system, you need to know its physical properties such as mass, stiffness, and damping. These properties determine how the system will respond to an external force.

When a force is applied to a system at a frequency equal to its natural frequency, the system absorbs energy from the force in maximum amounts during each cycle. This energy absorption leads to increased oscillation or vibration of the system, resulting in a larger amplitude.

It's important to note that resonance can lead to failure or damage in certain cases. For example, if a bridge is subjected to its natural frequency due to strong winds, it can lead to excessive vibrations, potentially causing structural damage. Resonance is also exploited in many technologies, such as musical instruments or wireless communication systems, to enhance performance.