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).

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

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.

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 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.

The fractional change in volume ΔV/V is given by ΔV/V = ΔP/B, where ΔP = 10 MPa and B = 200 GPa, resulting in 0.00005.

The change in volume ΔV/V = P/B, so ΔV/V = 50 MPa / 200 GPa = 0.025%.

Poisson's ratio is defined as the negative ratio of lateral strain to longitudinal strain, so a Poisson's ratio of 0.3 means the ratio is 0.3.

A material is considered elastic if it returns to its original shape after the removal of the applied stress.

Young's modulus (Y) = Stress / Strain = 100 MPa / 0.002 = 50 GPa.

A Poisson's ratio of 0.5 indicates that the material is incompressible.

Absolute error = Relative error * Measured value = 0.03 * 15.5 m = 0.465 m.

True value is taken as the average of the measured value and the error, which is (30 + 29.9) / 2 = 30.05 m.

Total uncertainty = 3 * 0.5 = 1.5 units.

Relative error = (absolute error / measured value) = 0.1 / 12.3 = 0.0081.

Relative error = (absolute error / measured value) = 0.1 / 5.0 = 0.02 or 2%.

Using conservation of energy: m1*c1*(T1-Tf) = m2*c2*(Tf-T2). Solving gives Tf = 50°C.

Work done = Power × Time = 1500 W × (10 × 60 s) = 90000 J.