Circular Motion Test. 3

Total linear acceleration is the vector sum of the tangential component (changing speed) and the centripetal component (changing direction).

Tension has a vertical component that balances weight and a horizontal component that provides the centripetal force.

At the top, both gravity and the normal force (if needed) act downwards, providing the centripetal force to keep the water moving in a circle.

For rotational equilibrium, there must be no net turning effect, meaning the sum of all torques is zero.

The negative potential energy is greater in magnitude than the positive kinetic energy, signifying a bound system.

At the equator, part of the gravitational force provides the centripetal force for rotation. The apparent weight is thus reduced: W_app = mg - mω²r.

Since τ = ΔL/Δt, then Δt = ΔL/τ = 10 J·s / 2 N·m = 5 s.

A spinning object has angular momentum. To change its direction (i.e., to make it fall), a torque is required. Rapid spinning makes it resistant to such torques.

The formula for orbital velocity is v_o = √(GM/R) and for escape velocity is v_e = √(2GM/R). Therefore, v_e = √2 v_o.

A geostationary orbit allows a satellite to remain over the same point on Earth's surface, which is ideal for continuous communication.

Moment of inertia is an intrinsic property of an object's mass distribution, independent of its motion.

This is the rotational equivalent of Newton's second law: τ = dL/dt.

Since one revolution is 2π radians, three revolutions is 3 × 2π = 6π radians.

Tangential acceleration acts along the tangent to the path, thus changing the object's speed.

Because the hoop has a larger moment of inertia, it will have a smaller translational velocity compared to the disk.

Moment of inertia I ∝ R². If R becomes R/2, I' becomes I/4. By conservation of angular momentum (Iω = I'ω'), ω' becomes 4ω. A faster rotation means a shorter day: T' = T/4 = 24/4 = 6 hours.

KE_rot = (1/2)Iω² = (1/5)Mv². KE_total = (7/10)Mv². The ratio is (1/5)/(7/10) = 2/7.

Artificial gravity is often measured in multiples of Earth's gravitational acceleration. '2g' means an acceleration of 2 × 9.8 m/s².

At the top, the tangential velocity (v) and the translational velocity (v) are in the same direction, so the resultant velocity is v + v = 2v.

Rotational equilibrium means the net torque is zero, so there is no change in angular velocity (i.e., zero angular acceleration).