Oscillations are a common phenomenon in our experience. Another example is the vertical oscillation of a mass suspended from the ceiling by a spring. A slightly more complicated example is the motion of a rocking chair, and a considerably more complicated example is the motion of a violin string. The common feature of these examples is that initially a system is somehow pushed away from its equilibrium configuration; however, the forces which act on the system drive it back toward the equilibrium configuration. When the system reaches the equilibrium configuration it has a finite velocity and therefore overshoots; once it is on the other side, the forces are again directed toward the equilibrium configuration to which it eventually returns, overshooting in the opposite direction and so on.
The motion of a particle under the influence of a force whose magnitude is proportional to the distance of the particle from its equilibrium position and whose direction is always toward the equilibrium position is called simple harmonic motion (SHM). Let us consider, for example, a particle moving in one dimension on a smooth horizontal surface. A spring is attached to the particle and the other end of the spring is attached to a wall as in Figure 1.
Let the equilibrium length of the spring be and let the actual length of the spring at a given instant be . It is assumed that in equilibrium the coils of the spring are partially open so that can be positive or negative. If is positive the spring exerts a force to the left and if is negative the spring exerts a force to the right. An “ideal” spring is one which obeys Hooke’s Law, which says that the magnitude of the force is proportional to the magnitude of . If we introduce a unit vector pointing away from the wall and let the force exerted on the particle by the spring be , then the quantitative statement of Hooke’s Law is
where is a constant called the spring constant. The minus sign in (1) ensures that if is positive (negative) the force is directed to the left (right).
Hooke’s Law (unlike Newton’s Laws) is not a fundamental law of nature, but most springs obey Hooke’s Law if is small enough. Every spring will deviate from Hooke’s Law if it is stretched or compressed too far. We shall assume that the magnitude of is small enough so that Hooke’s Law is valid. Since the acceleration of the particle is , Newton’s second law yields
Equation (2), plus the initial conditions (the initial position and velocity, i.e., the values of and at ), fully determines the subsequent motion. Mathematically, our problem is to find the function which satisfies (2) (which is called a “differential equation”) and takes on prescribed values for and at time .
To see that (2) plus specified initial values for and uniquely determine . we can imagine solving (2) numerically.
Let be a very small time increment and (for notational convenience) denoted by and denote by . Since and are known, we can compute and by using and , where (2) gives us the value of since we know . Now we know , , and (using (2)) . Now we can compute , , and by using and . Thus we can advance by small time increments. This procedure can be used to solve a second-order differential equation (an equation whose highest order derivative occurring is a second derivative) numerically even when we cannot get a solution in terms of familiar functions.
The mathematical problem can be solved by several different methods, which we shall now discuss. We rewrite (2) as
where ( is the lower-case Greek letter “omega”). A perfectly legitimate (though not very systematic) way to solve (3) is to guess at the solution and then to verify that the guess satisfies (3). In the preceding discussion we have already indicated that we expect to be an oscillating function of the time . The simplest oscillating function with which we are familiar is . Since and , we have . Thus we see that almost satisfies (3), up to the factor . This is easily fixed by trying the function . Since and we have . Thus we see that the function satisfies (3). Similarly we see that also satisfies (3). This is not surprising since the graph of is exactly the same as the graph of provided that the time origin is shifted; that is, . Finally,
satisfies (3) and is, in fact, its most general solution.
By choosing and appropriately we can arrange for and to take on any prescribed values at . Suppose that we want to have the value and to have the value at (physically we can let have any desired values if we start the motion “by hand” and then let go). Letting in (4) we find . Differentiating (4) with respect to time, we obtain Thus, the values of and which fit the initial conditions are and , and we obtain
Note that if the displacement is proportional to , and if the displacement is proportional to . Even when and we can, by an appropriate shift of the time origin, exhibit as a pure sine function or a pure cosine function. To do this we write (4) as
where it is understood that we always choose the positive square root. For given values of and there is a unique angle in the range such that
Note that (7) and (8) is consistent with the identity . From the graphs of and we see that
Using (7) and (8) we can rewrite (6) as
If we let then is proportional to . The change of variable from to corresponds to moving our time origin by an amount ; that is, when . The significance of is that it is the time when has its maximum positive value (since the cosine in (9) has the value when ). Thus we see that if we measure time from the instant when has its maximum positive value, the displacement is a pure cosine function. It is obvious from (6) or (9) that is periodic with period ; that is, . Therefore attains its maximum positive value not only at time , but also at times , , and so forth. The number of oscillations per second (the frequency) is . We also note that is called the angular frequency.
By now it should be clear that if we measure time from the instant when and is positive (i.e., the instant when the particle passes through its equilibrium position, traveling to the right) the displacement is a pure sine function. To prove this we note that and rewrite (9) as Thus we see that and when takes the following values
The maximum value of is called the amplitude of the oscillation. Inserting the calculated values of and we find
The equilibrium length of the spring is 1.00 meters and the mass is kg. The spring constant is N/m. At the mass is m from the wall and is given a velocity of m/s toward the left. Calculate (a) the angular frequency of the oscillation; (b) the period of the oscillation; (c) the amplitude of the oscillation; (d) the maximum and minimum distance of the mass from the wall; (e) the time when the mass is closest to the wall (calculate the smallest such time); (f) the time when the mass is furthest from the wall (calculate the smallest such time); (g) the time when the mass first is at a distance 1.10 m from the wall, traveling to the right; (h) the time when the mass first is at a distance m from the wall, traveling to the left.
The angular frequency is . The period is s. The amplitude is given by (10) Note that m since initially the mass is m to the left of its equilibrium position. Thus . The maximum distance of the mass from the wall is m. The minimum distance is m.
From (5) we have