Govt. Exams
For springs in series: 1/k_eff = 1/k + 1/k = 2/k, so k_eff = k/2
At highest point, for minimum speed, N = 0. mg = mv²/r, so v = √(gr)
Resultant force = √(3² + 4²) = √25 = 5 N. Mass = F/a = 5/2 = 2.5 kg
For rotating rod, energy: mg(L/2) = (1/2)Iω² + (1/2)mv². With I = mL²/3 about pivot, v_cm = √(gL/2)
Total acceleration = F/(m₁+m₂) = 10/5 = 2 m/s². Tension = F - m₁a = 10 - 2×2 = 6 N (or T = m₂a = 3×2 = 6 N)
a = g(sinθ - μcosθ) = 10(sin30° - 0.1×cos30°) = 10(0.5 - 0.1×0.866) = 10(0.5 - 0.0866) = 4.134 ≈ 4.3 m/s²
At maximum height, vertical component vᵧ = 0. Horizontal component vₓ remains 20 m/s. Total velocity = 20 m/s
Orbital velocity v = √(GM/r). If r → 2r, then v' = √(GM/2r) = v/√2
Tangential acceleration at = r(dω/dt) = 5 × 2 = 10 m/s² (constant as dω/dt = 2)
By momentum conservation: 4(5) - 6(3) = (4+6)v → 20 - 18 = 10v → v = 0.2 m/s