Chapter 10, Problem 139P

3.) 13.152* - A sphere (m = 2 kg) is connected to a fixed point O by an inextensible cord (L = 1.2 m). The sphere rests on a frictionless horizontal surface at a distance of x = 0.5 m from O when it is given a velocity (V.) in a direction perpendicular to OA. It moves freely until it reach A', when the cord becomes taut. Find the maximum possible velocity v. if the impulse of the force exerted on the cord is not to exceed F = 3Ns.

4.) 13.165* - Two billiard balls A & B (same mass m & diameter d = 2.37") are as shown. Ball A has velocity V₁ = 3 ft/s when it strikes B, which is at rest. B then moves in the x direction after impact. Find (a) angle 0, (b) the velocity of B after impact (VB'). Positional variables are x = 6" and y = 10", and e = 0.9.

We will consider a linear system and a nonlinear system under uncertainty, each expressed in the form of a set of stochastic differential equation (SDE) as follows: = da (Ax+ Bu)dt + Gdw, dx = f(x,u,t)dt+Gdw, (1) (2) where x is the state, u is the control, and dw is a differential increment of standard Brownian motion, i.e., E[dw] = 0 and E [dw(t)dw(t)] = dt-1. In this problem, we consider the linear SDE, Eq. (1), with a very simple model where x = R², u = [0,0] (no control), and dw R². The matrices A, B, and G are given as follows: A=02x2, B=02x2; G = [000] (3) where σp Є R represents the degree of the uncertainty, and let us take σ₁ = 2 and σ2 = 3. Assume that the initial state is deterministic and e(t = 0) = [0,0]. Take the following steps to simulate the given SDE for 1 € [0, 1]: (a): Consider the increments of w between each time interval [+1). Derive the analytical expression of Aw using w~N(02, 12), where Aw, w(tk+1) — w(tk). (c): Derive the approximate continuous-time EoM from…