Find the kinetic energy K of the block at the moment labeled B. (Express answer in terms of k and A)

MasteringPhysics: PS 12 Energy of Harmonic Oscillators Learning Goal: To learm to apply the law of conservation of energy to the analysis of harmonic oscillators. systems in simple harmonic motion, or harmonic oscillators obey the law of conservation of energy just like all other systems do. Using energy considerations, one can analyze m of of the such can if one assumes that mechanical en is not dissipated. In other words, E K+ U constant, where E is the total mechanical energy of the system, K is the kinetic energy, and U is the potential energy. As you know, a common example of a harmonic oscillator is a mass attached to a spring. In this problem, we will consider a moving block attached to a spring. Note that, since the gravitational potential energy is not changing in this case, it can be excluded from the calculations. For such a system, the potential energy is stored in the spring and is given by where k is the force constant of the spring and z is the distance from the equilibrium position. The kinetic energy of the system is, as always. 1mv2. where m is the mass of the block and vis the speed of the block. We will also assume that there are no resistive forces; that is, E constant. Consider a harmonic oscillator at four different moments. labeled A B. C and D. as shown in the figure. Assume that the force constant k, the mass of the block, m, and the amplitude of -A -A A/2 A vibrations, A, are given. Answer the following questions.

## Answer

We know that total energy TE = 0.5kA^2 Spring energy SPE = 0.5k (A/2)^2 kinetic energy K = TE -SPE = 0.5 kA^2 – 0.125 kA^2

= 0.375 kA^2 Answer