What is the initial acceleration of the block, a0a0, when the block is released at a distance AA to the right from its equilibrium position?

Question

Consider the system shown in the figure.(Figure 1) It consists of a block of mass mm attached to a spring of negligible mass and force constant kk. The block is free to move on a frictionless horizontal surface, while the left end of the spring is held fixed. When the spring is neither compressed nor stretched, the block is in equilibrium. If the spring is stretched, the block is displaced to the right and when it is released, a force acts on it to pull it back toward equilibrium. By the time the block has returned to the equilibrium position, it has picked up some kinetic energy, so it overshoots, stopping somewhere on the other side, where it is again pulled back toward equilibrium. As a result, the block moves back and forth from one side of the equilibrium position to the other, undergoing oscillations. Since we are ignoring friction (a good approximation to many cases), the mechanical energy of the system is conserved and the oscillations repeat themselves over and over.

The motion that we have just described is typical of most systems when they are displaced from equilibrium and experience a restoring force that tends to bring them back to their equilibrium position. The resulting oscillations take the name of periodic motion. An important example of periodic motion is simple harmonic motion (SHM) and we will use the mass-spring system described here to introduce some of its properties.

As shown in the figure(Figure 2), a coordinate system with the origin at the equilibrium position is chosen so that the x coordinate represents the displacement from the equilibrium position. (The positive direction is to the right.) What is the initial acceleration of the block, a0a0, when the block is released at a distance AA to the right from its equilibrium position?

Express your answer in terms of some or all of the variables AA, mm, and kk.

Answer 1

At distance A, the spring exerts force of magnitude kA on the mass.

By Newton's second law, such a force acting on mass m causes acceleration

kA/m