problem 1

A simple pendulum is idealized by considering a bob mass swinging on an inextensible, massless rod (negligible mass relative to the bob) about a frictionless pivot (Figure 1). Figure 1 Simple pendulum The pendulum oscillation is described by the initial-value problem consisting of the equation of motion with initial conditions: 8 + - sin(0) = 0, @(to) = 0, @(to) = wo Here, L is the length of the pendulum, g acceleration due to gravity, @ the angle the pendulum makes with the vertical, 0 the initial angular displacement, and wo the initial angular velocity. a) Transform the second-order initial-value problem into a system of first-order initial-value problems. b) Compute the angular displacement 0(0.2) and angular velocity @ (0.2) using the forward Euler method with step size h = 0.1. Use L = 0.5 m, g = 9.81 m/s', to = 0, 0. = = rad, and coo = 0 rad/s.The steady two-dimensional temperature (7) distribution in an isotropic heat conducting materials is given by Laplace equation, 7=0 The side lengths of the domain are _=8 and H=6. Assuming consistent units are used, boundary conditions are shown in Figure 2. Use the grid indicated in Figure 2 to solve for the temperature distribution. L =8 T=100 T=40 T=15 T-50 T1 H =6 y T=50 TA T=100 T=40 T=20 Figure 2 Finite difference nodal scheme Central finite difference formula f' ( x ) = f ( xth) - f (x-h) 2h f" ( x) = f ( xth) - 2f(x) +f(x-h) h2