FORCE LAW FOR SIMPLE HARMONIC MOTION -

COMBINATIONS OF SPRINGS

1. Springs in series
2. Springs in parallel

NOTE

For spring constant $\fn_cm \large \left [ k\propto\frac{1}{length\;of\;spring} \right ]$

FORCE LAW OF S.H.M

We know that if a body is subjected to a restoring force(F) that is proportional to its displacement from mean position (y) and is directed opposite to the direction of displacement. Then the body will execute S.H.M. This is the main reason for performing S.H.M.

i.e $\fn_cm \large F\propto y$

$\fn_cm \large F\propto -ky$

$\fn_cm \large ma=-ky$

$\fn_cm \large a=-\frac{k}{m}y$

$\fn_cm \large \left [ a=\omega^2 y \right ]\;\;\;\;\;where\;\;\omega^2=\frac{k}{m}$

$\fn_cm \large \left [ \omega=\sqrt{\frac{k}{m}} \right ]$

Once we find the value of force constant K of the system executing S.H.M, the time period and frequency of the system can be readily obtained.

$\fn_cm \large \because T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}}$

and $\fn_cm \large f=\frac{1}{T}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Now

$\fn_cm \large T=2\pi\sqrt{\frac{m}{k}}=2\pi\sqrt{\frac{my}{ky}}=2\pi\sqrt{\frac{my}{F}}=2\pi\sqrt{\frac{my}{ma}}$

$\fn_cm \large T=2\pi\sqrt{\frac{y}{a}}$

$\fn_cm \large \left [ T=2\pi\sqrt{\frac{displacement}{acceleration}} \right ]\;\;\;and\;\;\;\left [ f=\frac{1}{2\pi}\sqrt{\frac{acceleration}{displacement}} \right ]$

NOTE

If the force of particle is

$\fn_cm \large \left [ F=-kx^n \right ]$

(a) If n is an even integer (2,4,6,…..) then force is always along -ve x-direction for any value of x. Hence this is not in oscillation
(b) If n is an odd integer (1, 3, 5,…….), and if x>0 force is -ve, if x<0 force is +ve, if x=0 force is zero

i.e particle oscillate at x=0

(c) if n>1 oscillation is non-linear but if n=1 i.e F=-kx, particle oscillate in a linear way and called S.H.M

Basic differential equation of S.H.M

in SHM $\fn_cm \large F=-kx$

$\fn_cm \large \Rightarrow ma=-kx$

$\fn_cm \large m\frac{\mathrm{d^2 x} }{\mathrm{d} t^2}=-m\omega^2x\;\;\;\;\;(\because \omega=\sqrt{\frac{k}{m}})$

$\fn_cm \large \left [ \frac{\mathrm{d^2} x }{\mathrm{d} t^2}+\omega^2 x=0 \right ]$

The solution of this differential equation is sine or cosine function like

$\fn_cm \large x=A\sin(\omega t +\phi)\;or\;x=A\sin(\omega t +\phi)$

$\fn_cm \large \;or\;x=A\sin \omega t+B\cos \omega t$

Simple Harmonic Motion can be defined in two equivalent ways, either by $\fn_cm \large F=-kx\;\;\;or\;by\;\;\;y=A\sin(\omega t+\phi)$

Any function of t i.e x=f(t) oscillates in S.H.M if $\fn_cm \large \frac{\mathrm{d^2} x}{\mathrm{d} t^2}\propto x$ where x is displacement variable.

a. Two identical springs of spring constant k are attached to a block of mass m and to fixed supports as shown in Fig. 14.15. Show that when the mass is displaced from its equilibrium position on either side, it executes a simple harmonic motion. Find the period of oscillations.

b. A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body?

c. A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.

d. In the previous question let us take the position of mass when the spring is unstreched as x = 0, and the direction from left to right as the positive direction of the x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is

(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in
the amplitude or the initial phase?