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.

Damping ratio (ζ) = c / (2√(mk)) = 0.3 / (2√(1*4)) = 0.3 / 4 = 0.075.

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.

A damping ratio (ζ) < 1 indicates underdamped motion.

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.

Damped frequency (ω_d) = ω_n√(1-ζ^2) = 5√(1-0.2^2) = 5√(0.96) ≈ 4.8 rad/s.

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

The time period of a damped harmonic oscillator remains the same; damping affects amplitude, not period.

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.

Energy is proportional to the square of the amplitude, so if amplitude is doubled, energy increases by 2^2 = 4 times.

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.

Ratio = driving frequency / natural frequency = 2 / 1.5 = 1.33