The total energy E is conserved. When the displacement is half the amplitude, the potential energy is (1/2)E, so the kinetic energy is E - (1/2)E = (1/2)E.

Adding more soap decreases the surface tension because soap molecules disrupt the cohesive forces between water molecules.

The magnetic field inside a solenoid is given by B = μ₀nI.

The moment of inertia of a solid cylinder about its diameter is I = 1/4 MR^2.

For a solid cylinder, the total kinetic energy is KE_total = KE_translational + KE_rotational = (1/2)mv^2 + (1/2)(1/2)mR^2(ω^2). Since ω = v/R, the translational part is 2/3 of the total.

For a solid cylinder, the translational kinetic energy (KE_trans) is (1/2)mv² and the rotational kinetic energy (KE_rot) is (1/2)(Iω²). The ratio KE_trans:KE_rot is 1:2.

For a solid disk, the translational kinetic energy is 1/3 of the total kinetic energy when rolling without slipping.

The solid sphere will roll down the incline faster because it has a smaller moment of inertia compared to the hollow sphere.

Both will have the same speed at the bottom due to conservation of energy, as they start from the same height.

The moment of inertia of a solid sphere about an axis through its center is I = 2/5 MR^2.

The moment of inertia of a solid sphere about its center is I = 2/5 MR^2.

Using Raoult's Law, the mole fraction of the solvent can be calculated as 80 mmHg / 100 mmHg = 0.8, which corresponds to a mole fraction of 0.75 for the solvent.

The period (T) is the inverse of frequency (f); T = 1/f = 1/440 ≈ 0.00227 s.

Centripetal force (F_c) = mv²/r. Assuming mass (m) = 1 kg, F_c = 1(4²)/2 = 8 N.

Centripetal force = m(v²/r) = 1(4²/2) = 8 N. Tension in the string = 8 N.

Probability of guessing correctly = 1/4.

Probability of guessing correctly = 1/4.

Number of girls = 20 - 12 = 8. Probability = 8/20 = 2/5.

Probability = Number of boys / Total students = 12/20 = 3/5.

There are 20 - 8 = 12 boys in the group. The probability of selecting a boy is the number of boys divided by the total number of students, which is 12/20 = 3/5.

The probability of selecting a boy is the number of boys divided by the total number of students, which is 8/20 = 2/5.

The number of boys is 20 - 8 = 12. The probability of selecting a boy is 12/20 = 3/5.

Using the first law of thermodynamics, ΔU = Q - W = 100 J - 40 J = 60 J.

Using the First Law of Thermodynamics, ΔU = Q - W = 200 J - 50 J = 150 J.

Using the first law, ΔU = Q - W = 100 J - 150 J = -50 J.

Maximum possible temperature = Measured value + Error = 25 + 0.5 = 25.5°C.

Minimum temperature = Recorded temperature - Error = 25 - 0.5 = 24.5°C.

Maximum possible temperature = Measured value + Error = 100 + 0.5 = 100.5°C.

Force = Torque / Distance = 15 Nm / 0.3 m = 50 N.

Torque = Force × Distance × sin(θ) => 25 Nm = 10 N × 5 m × sin(θ) => sin(θ) = 0.5 => θ = 30 degrees.