Frequency (f) = 1/(2π)√(k/m) => k = (2πf)²m = (2π × 3)² × 0.5 ≈ 18 N/m.

Frequency (f) = (1/2π)√(k/m) => k = (2πf)² * m = (2π*3)² * 2 ≈ 18 N/m.

Period (T) = 1 / frequency = 1 / 2 Hz = 0.5 s.

Capacitance C is inversely proportional to the distance d between the plates. If d is halved, C doubles, so the new capacitance is 10μF.

Time period (T) = 2π√(L/g) = 2π√(1/9.8) ≈ 2 s

A plane mirror always produces a virtual and erect image, which is the same size as the object.

Using the formula: H = (u² * sin²θ) / (2g). H = (20² * (1/2)) / (2 * 9.8) = 10.2 m, approximately 10 m.

When a ray passes through the center of curvature, it reflects back on itself, forming a real and inverted image at the same distance as the object.

Using Snell's law, n1 * sin(i) = n2 * sin(r). Here, n1 = 1 (air), i = 30 degrees, n2 = 1.5 (glass). Therefore, sin(r) = (1 * sin(30))/1.5 = 0.333. Thus, r = sin^(-1)(0.333) ≈ 19.5 degrees, which is approximately 20 degrees.

According to the law of reflection, the angle of reflection is equal to the angle of incidence. Therefore, the angle of reflection is 30 degrees.

New angular momentum L' = I'ω' = (2I)(ω/2) = L.

The rotational kinetic energy is given by KE = 0.5 I ω² = 0.5 * 3 * (4)² = 24 J.

The rotational kinetic energy is given by KE = (1/2)Iω² = (1/2)(5 kg·m²)(4 rad/s)² = 40 J.

The solid cylinder has a lower moment of inertia, thus it accelerates faster and reaches the ground first.

The solid sphere has a smaller moment of inertia compared to the hollow sphere, allowing it to accelerate faster down the incline.

Using conservation of energy, potential energy at the top = kinetic energy at the bottom. The total kinetic energy is the sum of translational and rotational kinetic energy. Thus, v = √(5gh/7).

Using the formula λ = v/f, where v is the speed and f is the frequency, we have λ = 340 m/s / 1700 Hz = 0.2 m.

Potential Energy (PE) = 1/2 * k * x^2 = 1/2 * 200 N/m * (0.1 m)^2 = 1 J

Potential Energy in Spring (PE) = 1/2 kx^2 = 1/2 * 500 N/m * (0.2 m)^2 = 10 J

Using the equation of motion: h = ut + 0.5gt². Solving for t gives t ≈ 4 seconds.

Using τ = Iα, we have α = τ/I = 15/3 = 5 rad/s².

According to Newton's second law for rotation, τ = Iα. If τ is doubled, α also doubles.

Using the formula: final velocity = initial velocity + acceleration * time. Final velocity = 0 + 2 * 10 = 20 m/s.

Time period (T) = 1 / frequency = 1 / 440 Hz ≈ 0.00227 s

Time period (T) = 1 / frequency = 1 / 440 Hz ≈ 0.00227 s.

Using conservation of energy, potential energy at the top converts to rotational kinetic energy at the bottom. The angular speed ω = √(3g/L).

The speed of a wave is given by the formula v = fλ. Here, v = 50 Hz * 2 m = 100 m/s.

The speed of the wave can be calculated using the formula v = fλ. Here, v = 500 Hz * 0.5 m = 250 m/s.

The speed of the wave can be calculated using the formula v = fλ. Thus, v = 500 Hz * 0.68 m = 340 m/s.

Using the formula ω = ω₀ + αt, where ω₀ = 0, α = 2 rad/s², and t = 5 s, we get ω = 0 + 2*5 = 10 rad/s.