Figure 14.1 shows one of the simplest systems that can have periodic motion. A body with mass m rests on a frictionless horizontal guide system, such as a linear air track, so it can move along the x-axis only. The body is attached to a spring of negligible mass that can be either stretched or compressed. The left end of the spring is held fixed, and the right end is attached to the body. The spring force is the only horizontal force acting on the body; the vertical normal and gravitational forces always add to zero.
It’s simplest to define our coordinate system so that the origin O is at the equilibrium position, where the spring is neither stretched nor compressed. Then x is The x-component of the displacement of the body from equilibrium and is also the change in the length of the spring. The spring exerts a force on the body with x-component Fx, and the x-component of acceleration is ax = Fx/m.
Figure 14.2 shows the body for three different displacements of the spring. Whenever the body is displaced from its equilibrium position, the spring force tends to restore it to the equilibrium position. We call a force with this character a restoring force. Oscillation can occur only when there is a restoring force tending to return the system to equilibrium.
Let’s analyze how oscillation occurs in this system. If we displace the body to the right to x = A and then let go, the net force and the acceleration are to the left (Fig. 14.2a). The speed increases as the body approaches the equilibrium position O. When the body is at O, the net force acting on it is zero (Fig. 14.2b), but because of its motion it overshoots the equilibrium position. On the other side of the equilibrium position the body is still moving to the left, but the net force and the acceleration are to the right (Fig. 14.2c); hence the speed decreases until the body comes to a stop. We will show later that with an ideal spring, the stopping point is at x = -A. The body then accelerates to the right, overshoots equilibrium again, and stops at the starting point x = A, ready to repeat the whole process. The body is oscillating! If there is no friction or other force to remove mechanical energy from the system, this motion repeats forever; the restoring force perpetually draws the body back toward the equilibrium position, only to have the body overshoot time after time.
In different situations the force may depend on the displacement x from equilibrium in different ways. But oscillation always occurs if the force is a restoring force that tends to return the system to equilibrium