Govt. Exams
Centripetal and tangential accelerations are perpendicular. Total acceleration = √(a_c² + a_t²)
In SHM, v_max = ωA = (2π/T)A = 2πA/T
Initial ω = 100×2π/60 = 10π/3 rad/s. Final ω = 0. α = Δω/Δt = (10π/3)/10 = π/3 rad/s²
Net acceleration = g(sin θ - μ cos θ). Using v² = 2aL, v = √(2gL(sin θ - μ cos θ))
v_e = √(2GM/R). For new planet: v_e' = √(2G×8M/2R) = √(8GM/R) = 2√(2GM/R) = 2v_e
In elastic collision between equal masses where one is at rest, they exchange velocities. First ball stops (0 m/s), second moves at 10 m/s
a = (m₁ - m₂)g/(m₁ + m₂) = (3 - 2)×10/(3 + 2) = 10/5 = 2 m/s²
For precession, τ = dL/dt = Lω_p, so ω_p = τ/L
From torque balance about base: μ = cot(2θ)/2 = cot(120°)/2 = (1/√3)/2 ≈ 0.29, closest is 0.33 but calculation gives μ = 1/2√3 ≈ 0.29, however standard formula yields 0.5
PE_spring = (1/2)kx² = (1/2)×1000×(0.1)² = 5 J. At max velocity, 5 = (1/2)×2×v², v = √5 ≈ 2.24 m/s