= –( π s – ) × displacement = – ( π s – ) × (– . m) = m s – ⊳ . FORCE LAW FOR SIMPLE HARMONIC MOTION Using Newton’s second law of motion, and the expression for acceleration of a particle undergoing SHM (Eq. .
), the force acting on a particle of mass m in SHM is F ( t ) = ma = –m ω x ( t ) i.e., F ( t ) = –k x ( t ) ( . ) where k = m ω ( .14a) or ω = k ( .14b) Like acceleration, force is always directed towards the mean position—hence it is sometimes called the restoring force in SHM. To summarise the discussion so far, simple harmonic motion can be defined in two equivalent ways, either by Eq. ( .
) for displacement or by Eq. ( . ) that gives its force law. Going from Eq.
( . ) to Eq. ( . ) required us to differentiate two times.
Likewise, by integrating the force law Eq. ( . ) two times, we can get back Eq. ( .
). Note that the force in Eq. ( . ) is linearly proportional to x ( t ).
A particle oscillating under such a force is, therefore, calling a linear harmonic oscillator. In the real world, the force may contain small additional terms proportional to x , x , etc. These then are called non-linear oscillators. u Example .
Two identical springs of spring constant k are attached to a block of mass m and to fixed supports as shown in Fig. . . 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. Fig. . Displacement, velocity and acceleration of a particle in simple harmonic motion have the same period T, but they differ in phase u Example .
A body oscillates with SHM according to the equation (in SI units), x = cos [ π t + π / ]. At t = . s, calculate the (a) displacement, (b) speed and (c) acceleration of