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.

Mean = (10 + 12 + 14 + 16) / 4 = 13. Variance = [(10-13)² + (12-13)² + (14-13)² + (16-13)²] / 4 = 2.5.

Critical angle θc = sin^(-1)(1/n) = sin^(-1)(1/2.42) ≈ 24.4°.

Numbers greater than 4 are 5 and 6. Probability = 2/6 = 1/3.

The possible outcomes greater than 2 are {3, 4, 5, 6}. The even numbers among these are {4, 6}. Thus, the probability is 2/4 = 1/2.

Even numbers on a die: 2, 4, 6. Probability = 3/6 = 1/2.

The possible outcomes greater than 2 are {3, 4, 5, 6}. The even numbers among these are {4, 6}. Thus, the probability is 2/3.

The even numbers on a die are 2, 4, and 6, which gives us 3 favorable outcomes. The total outcomes are 6. Therefore, the probability is 3/6 = 1/2.

25% of 2000 calories = 0.25 * 2000 = 500 calories.

The dipole moment p = q * d.

The dipole moment p is defined as p = q * d.

The dipole moment (p) is defined as p = qd, where q is the charge and d is the separation distance.

Dipole moment p = q * d = (1 × 10^-6 C) * (0.1 m) = 1 × 10^-7 C m.

Dipole moment p = q * d, if d is doubled, p also doubles.

The torque (τ) experienced by a dipole in a uniform electric field is given by τ = pE sin(θ), where p is the dipole moment and θ is the angle between p and E.

Dipole moment p = q * d = (2 × 10^-6 C) * (0.1 m) = 2 × 10^-7 C m.

Torque τ = p × E = pE sin θ, where θ is the angle between p and E.

Angular momentum L = Iω, where I = (1/2)MR^2 for a disc.

Kinetic energy K = (1/2)Iω^2, where I = (1/2)MR^2 for a disc.

The moment of inertia I of a disc about its axis is (1/2)MR^2. Therefore, the kinetic energy K.E. = (1/2)Iω^2 = (1/2)(1/2)MR^2ω^2 = (1/4)MR^2ω^2.

The moment of inertia I of a disc about its axis is (1/2)MR². Therefore, angular momentum L = Iω = (1/2)MR²ω.

The moment of inertia of a solid disc about its central axis is given by I = (1/2)MR².

Rotational kinetic energy K.E. = (1/2)Iω². For a disc, I = (1/2)MR², thus K.E. = (1/4)MR²ω².

The relationship between linear speed and angular speed for rolling without slipping is ω = v/R.

Discount = 15% of $120 = $120 * 0.15 = $18. Final Price = $120 - $18 = $102.

Let the marked price be x. After a 15% discount, the selling price is 0.85x. Setting this equal to $255 gives 0.85x = $255, so x = $255 / 0.85 = $300.

Let the original price be x. Then, Selling Price = x - (25% of x) = 150 => 0.75x = 150 => x = 200.

The disk has a lower moment of inertia compared to the ring, thus it reaches the ground first.