Dynamics Flashcards

A 2 kg mass on a spring (k = 200 N/m) undergoes simple harmonic motion. What is its natural frequency ωn?

Answer: 10 rad/s

ωn = √(k/m) = √(200/2) = √100 = 10 rad/s.

A satellite in circular orbit at radius r from the Earth's center has speed v. If r doubles, what happens to v?

Answer: v decreases by factor √2

v = √(GM/r); doubling r gives v' = v/√2, a decrease by factor √2.

What is the velocity of the center of mass of a system of particles with masses m1=2 kg (v1=3 m/s) and m2=4 kg (v2=−1 m/s)?

Answer: 1/3 m/s

v_cm = (m1v1 + m2v2)/(m1+m2) = (6−4)/6 = 2/6 = 1/3 m/s.

D'Alembert's principle converts a dynamic problem into an equivalent static problem by introducing:

Answer: An inertia force −ma opposing acceleration

D'Alembert's principle adds the inertia vector −ma to the free-body diagram, yielding a pseudo-static equilibrium.

A gyroscope spinning at high speed has its spin axis horizontal. When a torque is applied vertically, the axis precesses. The precession rate Ω is given by:

Answer: Ω = M/(I·ω)

Precession rate Ω = M/(I·ω), where M is the applied torque, I is moment of inertia, and ω is spin rate.

A 3 kg block rests on a surface with μk = 0.2 and is pushed by a 20 N force. What is the acceleration? (g = 9.81 m/s²)

Answer: 4.75 m/s²

F_friction = μk·mg = 0.2×3×9.81 = 5.886 N; a = (20−5.886)/3 ≈ 4.71 m/s² ≈ 4.75 m/s².

For a critically damped single-degree-of-freedom system, the damping ratio ζ equals:

Answer: 1

Critical damping occurs exactly when ζ = 1, providing the fastest return to equilibrium without oscillation.