What is the acceleration a1a1 of the block when it passes through 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.

What is the acceleration a1a1 of the block when it passes through its equilibrium position?

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

Answer 1

The restoring force in case of simple harmonic motion is an elastic force of the spring. It could be found by the use of Hooke's law F = - kx. Here k - the spring constant and x - deformation of the spring (change of its length from equilibrium position).

We see that this is a variable force that depends on the displacement of the mass-spring system x. Minus sign tells us that the direction of this force is opposite to the displacement; as a result this force brings the mass back to the equilibrium position (this is why we call it a restoring force).

As any force, restoring force could be found using Newtons second law F = ma, where m - mass of the body and a - acceleration of the body.

Now we can write:

ma = -kx,

Then at any given time the acceleration of the body is

a = - (k/m) x

We see that it depends on x - displacement of the mass, i.e. also changing with time as displacement itself.

It means that it could be negative, positive or zero. Because acceleration is a vector, we could also conclude that the direction of the vector of acceleration changes with time.

When the body goes through equilibrium position its displacement x is zero. Hence, acceleration also 0.

BTW, the maximum value of the acceleration we will have when the amplitude of the mass-spring system is maximum. At this points the direction of the mass-spring pendulum switches to opposite. Those points we call inflection points.

Answer: 0