The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which can be expressed as (1/2)mv^2 + (1/2)(2/5)mr^2(ω^2).

The total kinetic energy of a rolling object is the sum of translational and rotational kinetic energy, which simplifies to (1/2)mv^2 + (1/4)mv^2 = (3/4)mv^2.

The linear velocity v of a rolling object is given by v = Rω.

For rolling without slipping, the relationship is v = ωR, where v is the translational speed and ω is the angular speed.

Angular momentum L = Iω, thus ω = L/I.

Angular momentum L = Iω = 4 kg·m² * 3 rad/s = 12 kg·m²/s.

Angular momentum L is given by L = Iω. Thus, L = 5 kg·m² * 3 rad/s = 15 kg·m²/s.

Angular momentum L = Iω = 5 kg·m² * 3 rad/s = 15 kg·m²/s.

The rotational kinetic energy is given by KE = 1/2 Iω^2.

A geostationary satellite has an orbital period equal to the Earth's rotation period, which is 24 hours.

In a stable orbit, the net force acting on the satellite is zero because the gravitational force provides the necessary centripetal force.

The potential energy of a satellite increases when it is moved to a higher orbit due to the increase in distance from the Earth's center.

The orbital period of a satellite increases when it is moved to a higher orbit, according to Kepler's third law.

For a satellite in a stable circular orbit, the centripetal acceleration is equal to the gravitational acceleration.

A satellite in a circular orbit possesses both kinetic energy due to its motion and potential energy due to its position in the gravitational field.

If a satellite's altitude is doubled, its orbital speed decreases by √2.

As the altitude increases, the orbital period increases due to the greater distance from the Earth's center.

If a satellite's speed exceeds the escape velocity, it will escape Earth's gravitational pull and move away into space.

If a satellite's speed is less than the required orbital speed, it will not have enough centripetal force to maintain its orbit and will fall back to Earth.

Time period (T) is the reciprocal of frequency (f). T = 1/f = 1/1 = 1 s.

The amplitude of a simple harmonic oscillator is defined as the maximum displacement from the equilibrium position, which is 5 cm in this case.

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.