7. Power Absorption

P(t)=F(t)⋅v(t)P(t) = F(t) \cdot v(t)

Instantaneous power is defined as the product of the force applied on an object and the velocity of that object at a given moment in time. It is a measure of how much work is done at that precise instant. In formulaic terms, it's expressed as P(t), where P represents power, F(t) is the force at time t, and v(t) is the velocity at that same time. This means if the force or the velocity changes, the power will also change. The unit of power is watts (W), which is equivalent to joules per second (J/s).

Think of driving a car. The power at any instant while accelerating depends on how hard you press the accelerator (force) and how fast the car is going (velocity). If you press down harder or speed up, the power increases.

In steady-state forced SHM: ⟨P⟩=12F0Acos δ⟨P⟩ = \frac{1}{2} F_0 A \cos \delta

Average power in a forced simple harmonic motion (SHM) scenario is calculated when conditions are steady, meaning that the system has reached a consistent pattern of motion. The average power ⟨P⟩ is given by the expression 1/2 times the product of the maximum force (F0) and the amplitude of motion (A), multiplied by the cosine of the phase difference (δ) between the driving force and the oscillation. This average gives us a more practical figure as it reflects the constant output of power over time, rather than fluctuations during rapid changes.

Imagine using a blender. At first, it may take time to reach a steady blending speed. Average power reflects the energy used to keep the blender running smoothly once it reaches that speed, rather than the power fluctuations during startup.

● At ω=ω0 delta=π2 delta = \frac{\pi}{2} ● Power absorbed is maximum Graph of power vs frequency is a Lorentzian curve, peaked at resonance

In this context, resonance occurs when the frequency of external periodic force (ω) matches the natural frequency (ω0) of the system. When this happens, the phase lag (δ) between the force and the resulting motion becomes π/2 radians, indicating that the force is maximally effective in doing work on the system. The power absorbed is at its peak during resonance, which means that the system can transfer energy most efficiently, leading to larger oscillations. The relationship between power and frequency is often illustrated as a Lorentzian curve, showing a sharp peak at resonance.

Think of pushing someone on a swing. If you push in time with the swing's natural motion, they go higher (resonance). If you push at the wrong time, your effort is less effective (lower power). This principle is crucial in designing systems like bridges and buildings to avoid resonance that can cause structural harm.

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