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

Torque (τ) = F × r × sin(θ) = 50 N × 1 m × sin(60°) = 50 N × 1 m × (√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.

Total marbles = 4 + 5 + 6 = 15. Probability of red or green = (4 + 5)/15 = 9/15 = 3/5.

Total marbles = 4 + 5 + 6 = 15. Non-green marbles = 4 + 6 = 10. Probability = 10/15 = 2/3.

Total marbles = 4 + 5 + 6 = 15. Non-blue marbles = 4 + 5 = 9. Probability = 9/15 = 3/5.

Total marbles = 4 + 5 + 6 = 15. Probability of green = 5/15 = 1/3.

The total number of marbles is 4 + 5 + 6 = 15. The probability of picking a green marble is 5/15 = 1/3.

The total number of marbles is 4 + 5 + 6 = 15. The probability of selecting a green marble is 5/15 = 1/3.

Total marbles = 5 + 3 + 2 = 10. Favorable outcomes (red or green) = 5 + 3 = 8. Probability = 8/10 = 4/5.

Total marbles = 5 + 3 + 2 = 10. Probability of green = 3/10.

The total number of marbles is 5 + 3 + 2 = 10. The number of favorable outcomes (red or green) is 5 + 3 = 8. Thus, the probability is 8/10 = 4/5.

Using tan(30°) = height/distance, we have 1/√3 = 100/distance. Therefore, distance = 100√3 ≈ 173.2 m.

Using tan(30°) = 100/distance, we have 1/√3 = 100/distance. Therefore, distance = 100√3 ≈ 173.21 m.

Using tan(60°) = height/distance, we have distance = height/tan(60°) = 30/√3 = 15√3 m.

Using tan(45°) = height/distance, we have 1 = 30/distance. Therefore, distance = 30 m.