Spring Vibrations

It is easy to make a ball oscillate by hanging it on a spring. Let us fasten one end of the spring and pull the ball (Figure 1). The spring will be in a stretched position as long as we pull the ball with our hand. If we let go, the spring will unstretch and the ball will begin moving towards its equilibrium position. Just as the pendulum, the spring will not come to a state of rest immediately. The equilibrium position will be passed by inertia, and the spring will begin compressing. The ball slows down, and at a certain instant it comes to a halt in order to start moving at once in the opposite direction. There arises an oscillation with the same typical features with which our study of the pendulum acquainted us.

In the absence of friction, the oscillation would continue indefinitely. In the presence of friction, the oscillations are damped; moreover, the greater the friction, the faster they are damped.

The roles of a spring and a pendulum are frequently analogous. Both one and the other serve to maintain constancy in the period of clocks. The precise movement of modern spring watches is ensured by the oscillatory motion of a small flywheel balance. It is set oscillating by a spring which winds and unwinds tens of thousands of times a day.

For the ball on a string, the role of the restoring force was played by the tangential component of gravity. For the ball on a spring, the restoring force is the elastic force of a contracted or stretched spring. Therefore, the magnitude of the elastic force is directly proportional to the displacement: .

The coefficient has another meaning in this case. It is now the stiffness of the spring. A stiff spring is one which is difficult to stretch or contract. The coefficient has precisely such a meaning. The following is clear from the formula: is equal to the force necessary for the stretching or contraction of the spring by a unit of length.

Knowing the stiffness of the spring and the mass of the load hung on it, we find the period of free oscillation with the aid of the formula . For example, a load of mass on a spring with a stiffness coefficient dyn/cm (this is a rather stiff spring-a hundred-gram weight will stretch it by 1  cm) will oscillate with a period . During one second, 16 oscillations will take place.

The more flexible the spring, the slower will be the vibration. An increase in the mass of the load has the same effect.

Let us apply the law of conservation of energy to a ball on a spring.

We know that the sum of the kinetic and potential energies, , for a pendulum does not vary:

is conserved

We know the values of and for a pendulum. The law of conservation of energy states that But the same thing is also true for a ball on a spring.

The deduction which we must inevitably make is quite interesting. Aside from the potential energy with which we became acquainted earlier, there also exists, therefore, a potential energy of a different kind. The former is called gravitational potential energy. If the spring were hanging horizontally, the gravitational potential energy would, of course, not change during the vibration. The new potential energy we discovered is called elastic potential energy. In our case it is equal to , i.e, it depends on the stiffness of the spring and is directly proportional to the square of the magnitude of contraction or stretching.

The total energy of the vibration, remaining constant, may be expressed in the following form: , or .

The quantities and occurring in the last formulas are the maximum values which the displacement and speed take on during the vibration. (They are sometimes called the amplitude values of the displacement and speed.) The origin of these formulas is quite clear. In an extreme position, when , the kinetic energy of vibration is equal to zero, and the total energy is equal to the potential energy. In the central position, the displacement of the point from the equilibrium position, and hence the potential energy, is equal to zero, the speed at this instant is maximum, and the total energy is equal to the kinetic energy.

The study of oscillations is an extensive branch of physics. One often has to deal with pendulums and springs. But this, of course, does not exhaust the list of bodies whose oscillations must be investigated. Mountings vibrate; bridges, parts of buildings, beams and high-voltage lines can begin vibrating. Sound is a vibration of the air.

We have listed several examples of mechanical vibrations. However, the concept of oscillation may refer not only to mechanical displacements of bodies or particles from an equilibrium position. We also come across oscillations in many electrical phenomena, moreover, these oscillations occur in accordance with laws closely resembling those which we have considered above. The study of oscillations permeates all branches of physics.