The capacitance increases by the factor of the dielectric constant, so C' = k * C, where k = 4.

Using F = ma, we find acceleration: a = F/m = 15 N / 3 kg = 5 m/s².

Work = Force × Distance = 15 N × 4 m = 60 J.

The temperature of the other end of the rod will increase over time due to conduction as heat is transferred along the rod.

A particle moving in a circular path with constant speed experiences centripetal acceleration directed towards the center of the circle.

Impulse is defined as the change in momentum of an object, which is calculated as the difference between the final and initial momentum, J = p2 - p1.

The person's speed relative to the ground is 60 km/h + 4 km/h = 64 km/h.

The plane's speed relative to the ground is the speed of the plane minus the speed of the wind: 200 km/h - 50 km/h = 150 km/h.

The time constant τ = R * C = 10 ohms * 100 x 10^-6 F = 0.001 seconds or 1 millisecond.

Using Ohm's Law, I = V/R. Here, I = 5V / 10Ω = 0.5 A.

R_total = R1 + R2 = 10 Ω + 20 Ω = 30 Ω.

Total resistance in series is R_total = R1 + R2 = 10 Ω + 5 Ω = 15 Ω.

Power P = V²/R = (20V)² / 10Ω = 400/10 = 40 W.

If no external torques act on the body, the angular velocity can remain constant during rotation.

The angular speed (ω) can be calculated using the formula ω = v/r. Here, ω = 2 m/s / 0.5 m = 4 rad/s.

The angular speed (ω) of a rolling object is related to its translational speed (v) by the equation ω = v/r.

Using KE = (1/2)Iω², we have 50 J = (1/2)(5 kg·m²)ω², solving gives ω = 10 rad/s.

Torque is calculated as τ = I * α, where I is the moment of inertia and α is the angular acceleration. Here, τ = 2 kg·m² * 3 rad/s² = 6 N·m.

Rotational kinetic energy is given by KE = 0.5 Iω² = 0.5 * 4 kg·m² * (3 rad/s)² = 18 J.

Using L = Iω, we find ω = L/I = 20 kg·m²/s / 4 kg·m² = 5 rad/s.

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

The observed frequency decreases as the source moves away, due to the Doppler effect.

The speed of sound increases when it travels from air (approximately 343 m/s) into water (approximately 1482 m/s).

According to the Doppler effect, if the source is moving towards the observer, the frequency increases.

Potential Energy = 0.5 × k × x² = 0.5 × 200 N/m × (0.5 m)² = 25 J.

The swimmer's speed relative to the bank when swimming upstream is 3 m/s - 1 m/s = 2 m/s.

Using τ = Iα, we have α = τ/I = 10 N·m / 2 kg·m² = 5 rad/s².

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.

If the angular velocity is constant, the angular acceleration is zero.

The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s².