Explanation:
Pendulum period T = 2π√(l/g) for small amplitudes. It depends on length l and gravity g, but not on bob mass or amplitude (isochronism for small angles). Mass independence arises because gravitational force and inertia both proportional to mass, canceling out. Memory aid: 'Pendulum: T ∝ √(l/g); independent of mass and small-amplitude'. This conceptual question tests oscillations fundamentals, frequently examined in competitive exams. Always verify the small-angle approximation; competitive exams assume it unless specified otherwise. This problem assesses understanding of which parameters affect periodic motion.
The escape velocity from Earth's surface is approximately 11.2 km/s. If a planet has twice Earth's radius and same density, its escape velocity will be:
Explanation:
Escape velocity v_e = √(2GM/R). Mass M = density ρ × volume = ρ×(4/3)πR³. Thus v_e = √(2Gρ×4/3πR³ / R) = √(8GρπR²/3) ∝ R√ρ. Given same density and R_planet = 2R_earth, v_e ∝ R, so v_e_planet = 2 × 11.2 = 22.4 km/s. This derivation shows escape velocity scales linearly with radius for constant density. Memory tip: v_e ∝ √(M/R) and M ∝ R³ for constant ρ, so v_e ∝ R. Such proportional reasoning questions test conceptual grasp of gravitation formulas in competitive exams without heavy calculation.
Explanation:
Conservative forces (gravity, spring, electrostatic) have path-independent work; work done over any closed loop is zero. This defines conservative forces and enables potential energy definition. Non-conservative forces (friction) have non-zero closed-path work. Memory aid: 'Conservative force: ∮F·dr = 0; enables energy conservation'. This mechanics concept is frequently tested in competitive exams. Always link conservative forces to potential energy and mechanical energy conservation; this is foundational for advanced physics problem-solving.
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