Torque (τ) = F × r × sin(θ) = 20 N × 0.5 m × sin(60°) = 20 N × 0.5 m × (√3/2) = 8.66 Nm.

Torque (τ) = F × r × sin(θ) = 20 N × 2 m × sin(60°) = 20 × 2 × (√3/2) = 20√3 Nm ≈ 34.64 Nm.

Work done = Force × Distance × cos(θ) = 25 N × 10 m × cos(30°) = 216.5 J.

Work done = Force × Distance × cos(θ) = 25 N × 3 m × cos(60°) = 37.5 J.

Torque (τ) = F × r × sin(θ) = 30 N × 2 m × sin(60°) = 30 × 2 × (√3/2) = 30√3 Nm ≈ 51.96 Nm.

Using F = ma, a = F/m = 30 N / 5 kg = 6 m/s².

Torque = Force × Distance = 40 N × 0.5 m = 20 Nm.

Torque (τ) = F × r × sin(θ) = 40 N × 2 m × sin(60°) = 40 × 2 × (√3/2) = 40√3 Nm ≈ 34.64 Nm.

Torque (τ) = F × r × sin(θ) = 50 N × 0.5 m × sin(60°) = 50 N × 0.5 m × (√3/2) = 43.3 Nm.

Torque (τ) = F × r × sin(θ) = 50 N × 2 m × sin(30°) = 50 N × 2 m × 0.5 = 50 N·m.

Torque (τ) = F × d × sin(θ) = 50 N × 1 m × sin(30°) = 50 N × 1 m × 0.5 = 25 Nm.

Torque (τ) = F × r × sin(θ) = 50 N × 1 m × sin(60°) = 50 N × 1 m × (√3/2) ≈ 43.3 Nm.

Torque (τ) = F × r × sin(θ) = 50 N × 2 m × sin(60°) = 50 × 2 × (√3/2) = 43.3 Nm.

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

The RMS speed is proportional to the square root of the temperature. Therefore, at 600 K, the RMS speed will be 400 * sqrt(600/300) = 400 * sqrt(2) m/s.

The RMS speed is proportional to the square root of the temperature. Therefore, v_rms at 600 K = 500 * sqrt(600/300) = 500 * sqrt(2) ≈ 707 m/s.

The work done in an isothermal expansion is W = nRT ln(Vf/Vi). Assuming 1 mole of gas, W = 1 * 8.31 * 300 * ln(2) ≈ 600 J.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T = (v_rms^2 * M) / (3R). Substituting v_rms = 500 m/s and M = 0.02 kg/mol gives T = 500 K.

Upper limit = Measured value + Error = 180 + 1 = 181 cm

For a hollow cylinder, the electric field at a point outside is E = λ/(2πε₀r) using Gauss's law.

The electric field inside a hollow charged sphere is zero due to symmetry.

Inside a hollow charged sphere, the electric field is zero due to symmetry.

The electric field inside a hollow charged sphere is zero due to symmetry.

For a hollow sphere, I = (2/3)mr^2. Using energy conservation, the translational kinetic energy is 2/3 of the potential energy at the top.

For a hollow sphere, the translational kinetic energy at the bottom is 2/3 of the total energy, hence the fraction is 2/3.

The moment of inertia of a hollow sphere about its center is I = (2/3)mR^2.

Inside a hollow charged sphere, the electric field is zero due to symmetry, as per Gauss's law.

Inside the cavity of a hollow conductor, the electric field is zero due to the shielding effect of the conductor.

Percentage error = (Absolute error / True value) * 100 = (1 / 100) * 100 = 1%.

Area = L², so uncertainty in area = 2 * L * (uncertainty in L) = 2 * 100 * 0.5 = 100 m².