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).

Tension T = mv²/r. If r is halved, T doubles for constant speed.

At the highest point, the centripetal force is provided by the weight of the mass, so T + mg = mv²/r. For T = 0, mg = mv²/r.

At the highest point, T + mg = mv²/r, thus T = mg - mv²/r.

At the highest point, the centripetal force is provided by the weight. Minimum speed = √(g*r).

At the bottom, T + mg = mv²/r; T = mv²/r - mg.

At the highest point, the centripetal force must equal the weight: mv²/L = mg, thus v = √(gL).

At the top, T + mg = mv²/L. T = mv²/L - mg.

The work done against gravity when lifting a mass m to a height h is given by W = mgh.

The frequency f of a mass-spring system is given by f = (1/2π)√(k/m). Here, f = (1/2π)√(200/0.5) ≈ 2 Hz.

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.

Angular frequency ω is related to frequency f by ω = 2πf. Therefore, for f = 1 Hz, ω = 2π(1) = 2π rad/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.

Period (T) = 1 / frequency = 1 / 2 Hz = 0.5 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.

Angular frequency ω = 2πf = 2π × 5 Hz = 10π rad/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.

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

Frequency f is the reciprocal of the period T. Thus, f = 1/T = 1/4 = 0.25 Hz.

The maximum speed v_max is given by v_max = Aω, where A is the amplitude and ω is the angular frequency. Thus, v_max = 0.1 * 20 = 2 m/s.

The fractional change in volume is given by ΔV/V = ΔP/B. Here, ΔP = 50 MPa = 0.05 GPa, so ΔV/V = 0.05/200 = 0.00025.