The period T is given by T = 2π√(m/k). Here, T = 2π√(2/200) = 2π√(0.01) = 2π(0.1) = 0.2π ≈ 0.63 s.

Angular frequency (ω) = 2πf = 2π(3) = 6 rad/s.

The period of a mass-spring system is T = 2π√(m/k). Halving the mass does not change the period since it is independent of mass in this case.

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.

The frequency of a mass-spring system is inversely proportional to the square root of the mass. Increasing the mass decreases the frequency.

Using the formula f = (1/2π)√(k/m), we can rearrange to find k = (2πf)²m. Thus, k = (2π(2))²(0.5) = 16 N/m.

Angular frequency (ω) = 2πf = 2π(3) = 6 rad/s.

Angular frequency ω = 2πf = 2π(3) = 6 rad/s.

The time period T is the reciprocal of frequency f. Thus, T = 1/f = 1/5 = 0.2 s.

Maximum speed v_max = ωA, where A is the amplitude. We have ω = √(k/m). Here, k = 100 N/m and m = 1 kg (assuming). Thus, A = v_max/ω = 4/10 = 0.4 m.

Angular frequency ω = 2π/T = 2π/2 = π rad/s ≈ 3.14 rad/s.

Frequency (f) is the reciprocal of the period (T). f = 1/T = 1/2 = 0.5 Hz.

The time period T of a mass-spring system in simple harmonic motion is given by T = 2π√(m/k).

The restoring force in simple harmonic motion is given by Hooke's law, which states that the force is proportional to the displacement and acts in the opposite direction. Therefore, the restoring force is -kx.

The angular frequency ω is given by ω = √(k/m).

The angular frequency ω is given by ω = 2πf. For f = 2 Hz, ω = 2π(2) = 4π rad/s.

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.

The angular frequency ω is related to the frequency f by the formula ω = 2πf. Therefore, ω = 2π × 2 = 4π rad/s.

The time period T is the reciprocal of frequency f. T = 1/f = 1/2 Hz = 0.5 s.

Angular frequency (ω) is given by ω = 2πf. Thus, ω = 2π(3) = 6 rad/s.

Angular frequency (ω) is given by ω = 2πf. Thus, ω = 2π * 3 ≈ 6 rad/s.

Period (T) is the reciprocal of frequency (f). T = 1/f = 1/3 = 0.33 s.

Period T = 1/f = 1/5 = 0.2 s.

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