A particle moves in a conservative force field where its potential energy U is given by U(x) = ax^2 - bx. The particle will be in stable equilibrium at position x equal to: MCQ with Answer and Explanation

Explanation:
While James Clerk Maxwell predicted electromagnetic waves mathematically, it was the German physicist Heinrich Hertz who successfully built the first spark-gap transmitter in 1887. He physically generated and detected radio waves in his laboratory, proving Maxwell's theories right. The SI unit of frequency, the Hertz (Hz), is named in his absolute honor.

Explanation:
Planck's constant (h) has dimensions [ML²T⁻¹]. Angular momentum (L = mvr) has dimensions [M][LT⁻¹][L] = [ML²T⁻¹], matching exactly. Linear momentum is [MLT⁻¹], force is [MLT⁻²], and torque is [ML²T⁻²]. This dimensional equivalence is fundamental in quantum mechanics, linking wave-particle duality. Memory tip: Both represent 'action' quantities with identical dimensional structure. This concept frequently appears in competitive exams testing dimensional analysis skills.

Explanation:
Though the ship is made of iron, it is hollow inside and traps a large volume of air. This makes the overall or average density of the ship less than the density of water. According to the law of floatation, an object floats if its average density is less than that of the fluid. A solid iron needle has a higher density than water, so it sinks.