A Spring with Mass. We usually ignore the kinetic energy of the moving coils of a spring, but lets try to get a reasonable approximation to this. Consider a spring of mass M, equilibrium length L0, and force constant k. The work done to stretch or compress the spring by a distance L is 1 2 kX2 , where X = L - L0. Consider a spring, as described above, that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. (a) Calculate the kinetic energy of the spring in terms of M and v. (Hint: Divide the spring into pieces of length dl; find the speed of each piece in terms of l, v, and L; find the mass of each piece in terms of dl, M, and L; and integrate from 0 to L. The result is not 1 2Mv2 , since not all of the spring moves with the same speed.) In a spring gun, a spring of mass 0.243 kg and force constant 3200 N>m is compressed 2.50 cm from its unstretched length. When the trigger is pulled, the spring pushes horizontally on a 0.053-kg ball. The work done by friction is negligible. Calculate the balls speed when the spring reaches its uncompressed length (b) ignoring the mass of the spring and (c) including, using the results of part (a), the mass of the spring. (d) In part (c), what is the final kinetic energy of the ball and of the spring?

Chapter 4 Notes There are natural and artificial sounds There are original and reproduced sounds Transient sound Occurs for a short period of time Source is set into vibration be a brief interaction and ends quickly after being created Steady sound Needs steady input of energy that maintains permanent vibration of sound source Going in pitch one octave higher will double the frequency Increasing the size of any instrument will lower its pitch Families of instruments: percussions, strings, pipes