The moment of inertia of a solid cylinder about its central axis is (1/2)MR^2.

For rolling without slipping, the relationship is v = Rω.

Using conservation of energy, potential energy at the top (mgh) converts to kinetic energy (1/2 mv^2 + 1/2 Iω^2). For a solid cylinder, I = 1/2 mr^2, leading to v = √(2gh).

Using conservation of energy, potential energy at height h converts to kinetic energy at the bottom. The speed is √(2gh).

The acceleration a = g sin(θ) / (1 + k^2/r^2) where k^2/r^2 for a solid cylinder is 1/2. Thus, a = g sin(θ) / (1 + 1/2) = g sin(θ)/2.

For rolling without slipping, a_cm = (2/3)g sin(θ).

Using the formula for surface area and volume, the optimal dimensions are found to be radius = 5 and height = 20.

Using the formula for surface area and volume, the optimal dimensions are found to be 5 cm height and approximately 15.87 cm radius.

Using the formula for volume and surface area, the optimal dimensions are found to be radius = 5 cm and height = 20 cm.

Resistivity is an intrinsic property of the material and does not change with geometry.

Resistance R = ρ * (L / A) = 2 x 10^-8 * (1 / (π * (0.01)²)) = 0.02 Ω.

For a point outside the cylinder, the electric field is given by E = λ/(2πε₀r).

The electric flux through the curved surface is proportional to the charge enclosed, which remains constant, so the flux through the curved surface doubles if the height is doubled.

The electric field due to a uniformly charged infinite cylinder depends only on the charge per unit length, not on the radius.

The electric flux through the curved surface of a cylinder is given by Φ = Q_enc/ε₀, where Q_enc = Q.

The electric field outside the cylindrical surface is directly proportional to the distance from the wire.

Gauss's law applies to electric fields, not magnetic fields. The electric field around a current-carrying wire is not defined by Gauss's law.

Using Gauss's law, the electric field outside the cylinder is E = Q/(2πε₀R).

Using Gauss's law, the electric field E at a distance R from the axis of a long charged cylinder is E = Q/(2πε₀L) for points outside the cylinder.

Stress is defined as force per unit area. Doubling the diameter increases the area by a factor of four, thus reducing the stress.

Tensile stress is given by force/area. Halving the radius reduces the area by a factor of four, thus the stress quadruples for the same force.

Total surface area = 2πr(h + r) = 2π(3)(5 + 3) = 2π(3)(8) = 48π ≈ 150.8 m².

The volume of a cylinder is given by the formula V = πr²h. Here, r = 3 and h = 5, so V = π(3)²(5) = 45π.

Volume = π * radius^2 * height = π * 3^2 * 5 = 45π cubic meters.

Resistance R = ρ(L/A) = 1.68 x 10^-8 * (1 / (π(0.5 x 10^-3)²)) = 0.0424 Ω.

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.

12C3 = 12! / (3!(12-3)!) = 220 combinations.

If the variance is 16, the standard deviation is 4. The minimum value can be calculated as 20 - 4 = 16. Therefore, range = 28 - 16 = 12.