Question

Exercise 3.2. This problem is challenging! Two identical particles of mass m are connected by a light spring with stiffness k (neglect the spring's mass) and equilibrium length 2. Ev- erything is lined up on the z-axis. Let the position of particle 1 be r(t) and the position of particle 2 be f(t). If at time t = 0, the positions are (0) = ? and r2(0) = l, and the velocities are non-zero with vi(0) = v1 +0 and v2(0) = 12 +0, determine the following: (a) The formulas describing the center-of-mass (COM) position X(t) and velocity V(t) as functions of time and the initial velocities vy, 02 and l.

Answer 1

initially at to AANME at any time to HAMAR H2(t) nilt) Position of center of mass at any time t is given by X(t) = m(n) + m (2) meni tu2) m + m 2 m X (t) = ni+U2 2 Differentiating with respect to time

de Cxwe) 1 5953 +(-3) V (t) = 1 al, (t) + 1 al (4) V (t) = alt) + 4 (4) (a (x (t) = V (t), velocity of com at any time to a (n.) = al, (t), velocity of first mass at any time +) a (12) = alt), velocity of second mass at any timet) initially at t zo V(o) = 9, 10) + V (0) - 20, +U₂ 2 2 V (0) = 4, + U now consider two mass Tystem as a single system. Spring force of two masses is internal system. applied force on of the

on this system no external force is applied, so the momentum of com of this system will not change, so we So we have Mxu (t) = constant where m-am, total mass of the system = mv (o) = m uct) (initial momentum = momentum at any timet, » V(o) = u(t) = a, tu 2 which means com of two masses moves with a constant velocity IV (t) = 4, + Uz 2 xta = initial position of com and now m(-1) + ml X(0) = mem

Position of com after any time to X (t) = 0 + displacement of com in timet X (t) = 0 + (a1, + 2) + (: displacement = velocity & time) alelocity of com= al, el Hent thence [X(t) = (ei teh ) + V (t) = a1, fU