Question

Problem 2. Recall that any undamped spring-mass system is described by an initial value problem of the form m" + ky= 0, (0) = 0, v(0) = to, where m is the mass and k is the spring constant. Since there is no damping, we would expect that no energy is lost as the mass moves. That is, the total energy (potential plus kinetic) in the system at any time I should equal the initial amount of energy in the system. "Actually, there's a little more to it than this. If ris repeated twice, then we just get" and te"and the third root will give another solution. If has multiplicity 3 (meaning it is repeated three times), we get solution of the form "te", and " (a) You may have seen in Calculus II, for example) that if F(y) represents a force as a function of position, the work done by the force in moving an object along a straight line from y = a to y = b is W = F(y) dy. By taking F to be the spring force F(y) = -ky, show that the potential energy initially stored in the spring is sky- (It will help to use the fact that the change in potential energy is W.) Conclude that the total energy (ie., potential plus kinetic energy) initially in the system is (ky + mea). (b) Write down the solution to the original initial value problem. (Your answer will be expressed in terms of m, k, To, and the (c) Using your solution from part (c), determine the total energy in the system at time L. (That is, find both the kinetic energy and potential energy in terms of t and add them together.) Your answer should match the initial total energy from part (b), thus confirming that total energy is conserved in an undamped spring-mass system.

Answer 1

Flyse -ky (spring force) tota so wa Scky) dy now, a - y10 = you bo ylt) = y so, ws. Il-kydy stored in - näky® + kys from this expression it is clear that at t=0, Potential energy stor system at 4:0 is á kyo the

now y'co) = Vo So Kinetic energy at t:o will be total energy : mvo+ Kyo? Hica Cryo + mv.3) eb) my" + ky=0 cof will be mg²+k=0 So, y, circostat + cz sin dat now at t=0, y = yo y = 1o. o so, yo) = Ci costo go intor so, ci =yo then y'.-c. yco): cala sin + cocos Ft - Vo => 4. Volem

So, Solution will be, 2) y' - -y. Esin Et + vo cost potential do at any time t we will have energy - hry *** (y? cos? E t + von Ch) Sin? Et + ayova cos | Etsin t'); Similary at any time. t 'we will have Kinetic energy thm cy'gz &m ( you want Sin? Et + va cost - 2 yo voy sin per cos ft) total energy at time it will be ky" + Imcy'st = $kyo?cos Tent + mvom sin sent + Yo vo mk Les (At sin het + kya Sin Senat + mv? los? Ent - yo vo mk sin so that cos that

á kyo ( sin sent + cos*Et) my? (Sin? Ft + cos Ft) 5 (kyo + mvo) this matches with the initial total energy. Thus we proved that total energy is conuerved in an undamped spring mass system.