Author: dexter

Problem Statement Simple pendulum (length l) in car accelerating horizontally at a. Find equilibrium angle and oscillation frequency. Given Information All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) Physical Concepts & Formulas This problem draws on fundamental physical principles. The key is to identify which…

Problem Statement Solve the Newton’s Laws / mechanics problem: Ball of mass m falls from height h onto spring (stiffness k). Find maximum compression x. All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) This problem draws on fundamental physical principles. The key is to identify…

Problem Statement Solve the momentum/collision problem: Solve the momentum/collision problem: Mass m₁ (velocity v₀) strikes stationary m₂ elastically. Find fraction of KE transferred. m₁ moving at v₀, m₂ at rest Elastic head-on collision $$v_2’=\frac{2m_1}{m_1+m_2}v_0,\quad \eta=\frac{\Delta T}{T_0}$$ Step 1: v₂’ = 2m₁v₀/(m₁+m₂). Step 2: ΔT = ½m₂v₂’² = 2 Given Information Masses $m_1$, $m_2$ and initial…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Particle of mass m in U = a/x + bx (a,b > 0). Find equilibrium position and small-oscillation frequency. U(x) = a/x + bx a,b > 0, mass m $$dU/dx=0\text{ at equilibrium},\quad \omega^2=\frac{1}{m}\frac{d^2U}{dx^2}\bigg|_{x_0}$$ Step 1: dU/dx = −a/x² + b = 0 → x₀ =…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Particle of mass m in U(x) = U₀(1−cos αx). Find force as function of x and frequency for small oscillations. U(x) = U₀(1−cos αx) Mass m $$F=-dU/dx,\quad \omega=\sqrt{k_{eff}/m}$$ Step 1: F = −dU/dx = −U₀α sinαx. Step 2: For small x: sinαx ≈ αx, so…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: Potential field U = a/r² (a > 0). Find force and determine if attractive or repulsive. U(r) = a/r² a = positive constant $$F_r=-\frac{dU}{dr}$$ Step 1: dU/dr = −2a/r³. Step 2: F_r = −dU/dr = +2a/r³. Step 3: F_r…

Problem Statement Solve the work-energy problem: Solve the work-energy problem: Force F = axî + bĵ. Find work done along path y = x² from (0,0) to (1,1). F = (ax, b) Path: y = x² $$W=\int(F_x dx+F_y dy)$$ Step 1: Parametrize: y = x², dy = 2x dx. Step 2: W = ∫₀¹(ax dx…

Problem Statement Solve the work-energy problem: Solve the work-energy problem: Mass m hauled distance l up rough incline (angle θ, friction μ) by force F parallel to incline. Find total work. Mass m, angle θ, coefficient μ Applied force F, distance l $$W_{total}=W_F+W_g+W_f$$ Step 1: W_F = Fl. Step 2: W_g = −mgl sinθ. Step…

Problem Statement Solve the Newton’s Laws / mechanics problem: Solve the Newton’s Laws / mechanics problem: Mass m on smooth incline (angle θ) attached to horizontal spring (stiffness k). Find equilibrium compression x. Mass m, angle θ Spring constant k, horizontal Along incline: mg sinθ = kx cosθ Step 1: Spring force component along incline…

Problem Statement Solve the work-energy problem: Solve the work-energy problem: Spring of stiffness k stretched by x₀, mass m released. Find KE when spring reaches natural length. Mass m, spring constant k, initial extension x₀ $$KE + PE = E_{total} = \tfrac{1}{2}kx_0^2$$ Step 1: Initial energy: E = ½kx₀² (all PE). Step 2: At natural…