Damping ratio (ζ) = c / (2√(mk)) = 0.5 / (2√(2*8)) = 0.5 / (2√16) = 0.5 / 8 = 0.0625.

Amplitude after time t = A0 * e^(-t/τ) = 10 * e^(-6/3) = 10 * e^(-2) ≈ 2.5 m.

Time constant (τ) = m/c, thus c = m/τ = 1/3 = 0.333 kg/s. Using c = 2ζ√(mk), we find ζ = 0.5.

Using F = mAω², we find A = F / (mω²) = 12 / (3*(2π*2)²) ≈ 0.2 m.

Maximum velocity (v_max) = Aω = A(2πf) = 0.1 * (2π * 2) = 0.4 m/s.

Maximum velocity (v_max) = Aω, where ω = 2πf. Assuming f = 1 Hz, v_max = 0.1 * 2π * 1 = 0.2 m/s.

In a damped system, the damped frequency is always less than the natural frequency.

New frequency (ω_d) = ω_n√(1-ζ²) = 3√(1-0.1²) ≈ 2.9 Hz.

The damped frequency is less than the natural frequency due to the effect of damping.

Natural frequency (ω_n) = √(k/m) = √(20/5) = √4 = 2 Hz.

At frequencies much lower than the natural frequency, the amplitude of the forced oscillator increases significantly.

Using the formula A(t) = A_0 e^(-ζω_nt), we find ζ = 0.2.

A damping ratio greater than 1 indicates overdamped behavior in the system.

The damping ratio can be calculated using the logarithmic decrement method, leading to ζ = 0.25.

Damping ratio (ζ) = c / (2√(mk)). If m is doubled, ζ is halved.

The damping coefficient determines how quickly the energy is lost in a damped harmonic oscillator.

The damping coefficient determines the rate of energy loss in a damped harmonic oscillator.

The damping coefficient determines how quickly energy is lost in a damped harmonic oscillator.

Using E(t) = E_0 e^(-2ζω_nt), we find ζ = 0.2.

When the driving frequency equals the natural frequency, resonance occurs, leading to maximum amplitude.

Ratio = driving frequency / natural frequency = 5 Hz / 4 Hz = 1.25.

Increasing the amplitude of the driving force generally increases the amplitude of the oscillation in a forced system.

Maximum amplitude occurs when the driving frequency is equal to the natural frequency.

In forced oscillations, the amplitude is directly proportional to the driving force in the steady state.

When the driving frequency matches the natural frequency, resonance occurs, leading to maximum amplitude.

When the driving frequency matches the natural frequency, resonance occurs, leading to maximum amplitude.

Increasing the amplitude of the driving force generally increases the amplitude of the forced oscillation.

At resonance, the driving frequency matches the natural frequency of the system, leading to a significant increase in amplitude.

The maximum amplitude achieved at resonance is referred to as the resonance peak.

Increasing the amplitude of the driving force generally increases the amplitude of the forced oscillation.