Question

Problem 1 A block of mass m is sliding on a frictionless, horizontal surface, with a velocity vi . It hits an ideal spring, of spring constant k, which is attached to the wall. The spring compresses until the block momentarily stops, and then starts expanding again, so the block ultimately bounces off (see Example 5.6.2). (a) Write down an equation of motion (a function x(t)) for the block, which is valid for as long as it is in contact with the spring. For simplicity, assume the block is initially moving to the right, take the time when it first makes contact with the spring to be t = 0, and let the position of the block at that time to be x = 0. Make sure that you express any constants in your equation (such as A or ω) in terms of the given data, namely, m, vi , and k. (b) Sketch the function x(t) for the relevant time interval.

Answer 1

Let X(t)= A Sin (wt+d) be the position of of the block. A= amplitude, w= angular frequency X(0)= 0 [The position at tro] : 0= A Sinco + $) 1 Sing=o = $=0 i. x(t) = A sincwt) , wala

using conservation of energy to find the amplitude A. Okt 20 =0 - v r=0 when X=A Ok= change in kinetic energy = o- Imv? OK = 1 mu? GU=1&2 -0 =1 / KA² oktav = - mu? + 4k4 =0 = A=/ : (x2+)(Te vi) sin( JE +)) The position of mass at . time t 2

- AX(t) X(t) vs t graph T22 1