Simple Harmonic Motion vs other analogous problems

Question

I've reading about Simple harmonic motion, and it is said that SHM occurs only when there is force $F \propto -kx$. This is true for spring-mass systems. However, in systems like the Pendulum, we have the torque $\tau$ which is proportional to angular displacement, such that $\tau \propto -\kappa\theta$.

So, even though it is the angular displacement $\theta$, instead of $x$, that is undergoing sinusoidal motion, is it okay to call this motion 'Simple Harmonic' ? If so, then shouldn't it be wrong to say that SHM occurs only when 'Force' is inversely proportional to 'Distance'

For example, in Series and Parallel LC circuits, you have oscillations of charge in the capacitor, that has a sinusoidal form exactly analogous to the displacement $x$ in a spring mass system. However, we don't call this SHM.

When exactly is the terminology $SHM$ applicable when describing an oscillatory system ? If one says, that it is only applicable, if the oscillating variable is the displacement, then shouldn't pendulum or torsional pendulum be disqualified from being labeled as $SHM$, since it is the angular displacement that oscillates in their case, instead of the ordinary displacement $x$?

However, I'm also led to believe that torque is generated by force, so in some sense force is more fundamental than torque. In a torsional pendulum, does $\tau\propto-\kappa\theta$ automatically imply some $F\propto -kx$ ? If so, where is this force coming from, and can we analyze a torsional pendulum, purely using forces without using any kind of torque?

Is SHM just one of the analogous systems that can be modeled by the more general Harmonic oscillator terminology?

Answer 1

The mathematical representation of the particle in simple harmonic motion model is based on three separate equations, not only one (which you have pointed out): $$F = -kx$$ $$\frac{d^2x}{dt^2} = -\omega^2x$$ $$x(t) = A\text{cos}(\omega t+\phi)$$

Hence if you are analyzing a mechanical situation and you find that it is of the form of any of these equations, it's motion can be modelled as simple harmonic.

In the case of the torsional pendulum, it's position can be described as: $$\tau = I\frac{d^2\theta}{dt^2} = -\kappa\theta$$ which is analogous to the second equation and the situation is a mechanical one. Hence, as long as the elastic limits of the wire isn't exceeded, this motion can be modelled as simple harmonic.

An LC circuit cannot be strictly modelled as simple harmonic, because it is not a mechanical one. However, the oscillations of the value of electric and magnetic field in the capacitor and inductor very closely resemble a spring-block system. This fact is used to obtain the equation: $$q = Q_{\text{max}} \text{cos}(\omega t + \phi)$$

Also note

'Force' is inversely proportional to 'Distance'

The term $x$ here refers to position, relative to the equilibrium position, and not strictly the distance.

Hope this helps.

Answer 2

I think that most physicists use simple harmonic to describe a variation in some quantity, $\xi$, (be it linear displacement, angular displacement or even electric current) that is sinusoidal with respect to time. In other words $$\xi=\xi_0 \sin(\omega t - \phi)$$ in which $\xi_0$, $\omega$ and $\phi$ are constants.

Alternatively we could say that $\xi$ has to be a real quantity obeying the equation $$\frac{d^2 \xi}{dt^2}=-\omega^2 \xi$$ A periodic but non-sinusoidal variation would, on account of Fourier's theorem, be a "compound harmonic motion", though the term is seldom used.

Answer 3

In the case of the simple pendulum, this is usually referred to as approximately simple harmonic motion, and is only regarded as valid for small angles of deflection. This is because the equation of motion is $$l\ddot{\theta}=-g\sin\theta$$ and when $\theta$ is sufficiently small, we can make the approximation $$\sin\theta\simeq\theta,$$thus giving the SHM equation in its standard form.