# Force Law for Simple Harmonic Motion

Once we know how the acceleration of a particle varies with time, we can use Newtons second law to learn what force must act on the particle to give it that acceleration. If we combine Newton's second law and acceleration amplitude, we find, for simple harmonic motion,a(t) = - Ï‰

^{2 }x(t) --------- (i)

F = ma = - (mÏ‰

^{2})x --------- (ii)

This results in a force proportional to the displacement but opposite in sign. It is Hooke's laws,

F = - kx, -------- (iii)

For a spring, the spring constant here being

k = mÏ‰

^{2}--------(iv)

We can in fact take Equation (iii) as an alternative definition of simple harmonic motion. It says:

The block-spring system of figure forms a linear simple harmonic oscillator (linear oscillator, for short), where "linear" indicates that F is proportional to x rather than to some other power of x. The angular frequency Ï‰ of the simple harmonic motion of the block is related to the spring constant k and the mass m of the block by Equation (iv), which yields.

Ï‰ = (angular frequency) --------- (v)

Ï‰ = --------- (vi)

By combining Equations (v) and (vi), we can write, for the period of the linear oscillator of figure

T = 2Ï€

^{}(period) --------- (vii)

Equations (v) and (vi) tell us that a large angular frequency (and thus a small period) goes with a stiff spring (large k) and a light block (small m).

Every oscillating system, be it the linear oscillator of figure, a diving board, or a violin spring string, has some element of "springiness" and some element of "inertia", or mass, and thus resembles a linear oscillator.

In the linear oscillator of figure, these elements are located in separate parts of the system, the springiness being entirely in the spring, which we assume to be massless, and the inertia being entirely in the block, which we assume to be rigid. In a violin string, however, the two elements are both within the string itself.

# Energy in simple harmonic motion

The potential energy of a linear oscillator like that of spring-mass system is associated entirely with the spring. Its value depends on how much the spring is stretched or compressed, that is, on x (t).U (t) = = ------------ (viii)

^{2}A (as here) is the same as (cos A)

^{2}but is not the same as cos A

^{2}, which means cos (A

^{2}).

The kinetic energy of the system is associated entirely with the motion of the block. Its value depends on how fast the block is moving, that is, on v (t) (velocity).

We know that velocity is given by

v(t) = -Ï‰ x

_{m}sin(Ï‰ t +Ï• ) ------------ (ix)

K(t) = ------------ (x)

If we use equation 14.6 to substitute k/m for Ï‰

^{2}, we can write Equation (x) as

K(t) = ------------- (xi)

The total energy is given by

E = U +K

=

=

For any angle Î±,

cos

^{2}Î± + sin

^{2}Î± = 1

Thus the quantity in the square brackets above is unity and we have

E = U + K =

The mechanical energy of a linear oscillator is indeed a constant, independent of time. The potential energy and kinetic energy of the linear oscillator are shown as functions of time in Figure (a), and as functions of displacement in Figure (b).

You might now understand why an oscillating system normally contains an element of springiness and an element of inertia: it uses the former to store its potential energy and the latter to store its kinetic energy.

Figure (a): The potential energy U(t), kinetic energy K(t), and mechanical energy E as functions of time, for a linear harmonic oscillator. Note that all energies are positive and that the potential energy and the kinetic energy peak twice during every period.

Figure (b) The potential energy U(x), kinetic energy K(x), and mechanical energy E as functions of position, for a linear harmonic oscillator with amplitude x

_{m}. For x = 0 the energy is all kinetic, and for x = Â± x_{m}it is all potential.