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Simple Harmonic Oscillation:
A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists
of a mass m, which experiences a single force F, which pulls the mass in the direction
of the point x = 0 and depends only on the position x of the mass and a constant k.
Balance of forces ( Newton 's second law) for the system is
Simple harmonic oscillations
Consider a mass m held in an equilibrium position by springs, as
shown in Figure 2A. The mass may be perturbed by displacing it to the
right or left. If x is the displacement of the mass from equilibrium
(Figure 2B), the springs exert a force F proportional to x, such that
mass held by springs
Figure 2: (A) A mass m held in equilibrium by springs. (B) A mass m displaced a distance x.
where k is a constant that depends on the stiffness of the springs.
Equation (10) is called Hooke’s law, and the force is called the spring
force. If x is positive (displacement to the right), the resulting force is
negative (to the left), and vice versa. In other words, the spring force
always acts so as to restore mass back toward its equilibrium position.
Moreover, the force will produce an acceleration along the x direction
given by a = d2x/dt2. Thus, Newton’s second law, F = ma, is applied to
this case by substituting −kx for F and d2x/dt2 for a, giving
−kx = m(d2x/dt2). Transposing and dividing by m yields the equation
Equation (11) gives the derivative—in this case the second derivative—
of a quantity x in terms of the quantity itself. Such an equation is
called a differential equation, meaning an equation containing
derivatives. Much of the ordinary, day-to-day work of
theoretical physics consists of solving differential equations. The
question is, given equation (11), how does x depend on time?
The answer is suggested by experience. If the mass is displaced and
released, it will oscillate back and forth about its equilibrium position.
That is, x should be an oscillating function of t, such as a sine wave or
a cosine wave. For example, x might obey a behaviour such as
Equation (12) describes the behaviour sketched graphically in Figure
3. The mass is initially displaced a distance x = A and released at
time t = 0. As time goes on, the mass oscillates from A to −A and back
to A again in the time it takes ωt to advance by 2π. This time is
called T, the period of oscillation, so that ωT = 2π, or T = 2π/ω.
The reciprocal of the period, or the frequency f, in oscillations per
second, is given by f = 1/T = ω/2π. The quantity ω is called the angular
frequency and is expressed in radians per second.
simple harmonic motion
Figure 3: The function x = A cos ωt.
The choice of equation (12) as a possible kind of behaviour satisfying
the differential equation (11) can be tested by substituting it into
equation (11). The first derivative of x with respect to t is
Therefore, V = - ωA Sin ωt.
Differentiating a second time gives
Therefore, a = - ω2 x