Simple Harmonic Oscillators SHO

Figure 1.1:

We consider the spring-mass system shown in Figure 1.1. A massless spring, one of whose ends is fixed has its other attached to a particle of mass $m$ which is free to move. We choose the origin $x=0$ for the particle's motion at the position where the spring is unstretched. The particle is in stable equilibrium at this position and it will continue to remain there if left at rest. We are interested in a situation where the particle is disturbed from equilibrium. The particle experiences a restoring force from the spring if it is either stretched or compressed. The spring is assumed to be elastic which means that it follows Hooke's law where the force is proportional to the displacement $F=-k x$ with spring constant $k$.

The particle's equation of motion is

$$m \frac{d^2 x}{d t^2}=- k x$$ (1.1)

which can be written as

$$\ddot{x}+ \omega_0^2 x =0$$ (1.2)

where the dots $\ddot{}$ denote time derivatives and

$$\omega_0=\sqrt{\frac{k}{m}}$$ (1.3)

It is straightforward to check that

$$x(t)=A \cos (\omega_0 t + \phi)$$ (1.4)

is a solution to eq. (1.4).

We see that the particle performs sinusoidal oscillations around the equilibrium position when it is disturbed from equilibrium. The angular frequency $\omega_0$ of the oscillation depends on the intrinsic properties of the oscillator. It determines the time period

and the frequency $\nu = 1/T$ of the oscillation. Figure 1.2 shows oscillations for two different values of $\omega_0$.
Problem 1: What are the values of $\omega_0$ for the oscillations shown in Figure 1.2? What are the corresponding spring constant $k$ values if ?
Solution: For A and $k=(2 \pi)^2 \, {\rm N m^{-1}}$; For B $\omega_0=3 \pi \, s^{-1}$ and $k=(3 \pi)^2 \, {\rm N m^{-1}}$

Figure 1.2:

The amplitude $A$ and phase $\phi$ are determined by the initial conditions. Two initial conditions are needed to completely specify a solution. This follows from the fact that the governing equation (1.2) is a second order differential equation. The initial conditions can be specified in a variety of ways, fixing the values of $x(t)$ and $\dot{x}(t)$ at $t=0$ is a possibility. Figure 1.3 shows oscillations with different amplitudes and phases.
Problem 2: What are the amplitude and phase of the oscillations shown in Figure 1.3?
Solution: For C, A=1 and $\phi=\pi/3$; For D, A=1 and $\phi=0$; For E, A=1.5 and $\phi=0$;

Figure 1.3: