1.3 Linear shm

The subject matter of this module is linear SHM – harmonic motion along a straight line about the point of oscillation. There are various physical quantities associated with simple harmonic motion. Here, we intend to have a closer look at quantities associated with SHM like velocity, acceleration, work done, kinetic energy, potential energy and mechanical energy etc. For the sake of completeness, we shall also have a recap of concepts already discussed in earlier modules.

The SHM force relation “F = -kx” is a generic form of equation for linear SHM – not specific to block-spring system. In the case of block-spring system, “k” is the spring constant. This point is clarified to emphasize that relations that we shall be developing in this module applies to all linear SHM and not to a specific case.

Since displacement of SHM can be represented either in cosine or sine forms, depending where we start observing motion at t = 0. For someone, it is easier to visualize beginning of SHM, when particle is released from positive extreme. On the other hand, expression in sine form is convenient as particle is at the center of oscillation at t = 0. For this reason, some prefer sine representation.

The very fact that there are two ways to represent displacement may pose certain ambiguity or uncertainty in mind. We shall , therefore, strive to maintain complete independence of forms with the understanding that when it is cosine function, then starting reference is positive extreme and if it is sine function, then starting reference is center of oscillation. In order to illustrate flexibility, we shall be using “sine” expression of displacement in this module instead of cosine function, which has so far been used.

Displacement

The displacement of the particle is given by :

$$x=A\sin \left(\omega t+\phi \right)$$

where “A” is the amplitude,"ω" is angular frequency, “φ” is the phase constant and “ωt + φ” is the phase. Clearly, displacement is periodic with respect to time as it is represented by bounded trigonometric function. The displacement “x” varies between “-A” and “A”.

Velocity

The velocity of the particle as obtained from the solution of SHM equation is given by :

$$v=\omega \sqrt{\left(A^2-x^2\right)}$$

This is the relation of velocity of the particle with respect to displacement along the path of oscillation, bounded between “-ωA” and “ωA”. We can obtain a relation of velocity with respect to time by substituting expression of displacement “x” in the above equation :

$$\Rightarrow v=\omega \sqrt{\left(A^2-x^2\right)}=\omega \sqrt{\{A^2-A^2\sin {}^2\left(\omega t+\phi \right)\}}=\omega A\cos \left(\omega t+\phi \right)$$

We can ,alternatively, deduce this expression by differentiating displacement, “x”, with respect to time :

$$\Rightarrow v=\frac{dx}{dt}=\frac{d}{dt}A\sin \left(\omega t+\phi \right)=\omega A\cos \left(\omega t+\phi \right)$$

The variation of velocity with respect to time is sinusoidal and hence periodic. Here, we draw both displacement and velocity plots with respect to time in order to compare how velocity varies as particle is at different positions.

The upper figure is displacement – time plot, whereas lower figure is velocity – time plot. We observe following important points about variation of velocity :