Define forced oscillation and resonance.

Forced oscillation is the motion of a system that is driven by an external force. Resonance occurs when the frequency of the external force matches the natural frequency of the system, causing a large amplitude response.

Forced oscillation is a type of motion where a system is driven by an external force. This force can be periodic or non-periodic, and can be applied to any system that has a natural frequency of oscillation. The motion of the system is described by a differential equation of the form:

m(d^2x/dt^2) + c(dx/dt) + kx = F(t)

where m is the mass of the system, c is the damping coefficient, k is the spring constant, x is the displacement of the system from its equilibrium position, and F(t) is the external force.

Resonance occurs when the frequency of the external force matches the natural frequency of the system. This causes a large amplitude response, as the system is able to absorb energy from the external force. The natural frequency of a system is given by:

ω = sqrt(k/m)

where ω is the angular frequency of the system.

Resonance can be beneficial or detrimental, depending on the application. In some cases, resonance can be used to amplify signals or increase the efficiency of a system. In other cases, resonance can cause damage to the system or lead to instability.

Overall, forced oscillation and resonance are important concepts in the study of oscillatory motion. Understanding these concepts can help us to design and analyse systems that exhibit oscillatory behaviour.