the projection of the velocity v of the reference particle, P. Fig. . The acceleration, a(t), of the particle P ′ is the projection of the acceleration a of the reference particle P.
Eq. ( . ) gives the acceleration of a particle in SHM. The same equation can again be obtained directly by differentiating velocity v ( t ) given by Eq.
( . ) with respect to time: d ( ) ( ) d a t = v t ( . ) We note from Eq. ( .
) the important property that acceleration of a particle in SHM is proportional to displacement. For x( t ) > , a ( t ) < and for x ( t ) < , a ( t ) > . Thus, whatever the value of x between – A and A , the acceleration a ( t ) is always directed towards the centre. For simplicity, let us put φ = and write the expression for x ( t ), v ( t ) and a ( t ) x ( t ) = A cos ω t , v ( t ) = – ω Asin ω t , a ( t )=– ω A cos ω t The corresponding plots are shown in Fig.
. . All quantities vary sinusoidally with time; only their maxima differ and the different plots differ in phase. x varies between – A to A ; v ( t ) varies from – ω A to ω A and a ( t ) from – ω A to ω A.
With respect to displacement plot, velocity plot has a phase difference of π / and acceleration plot has a phase difference of π . Using Eq. ( . ), the speed of the body = – ( .
m)( π s – ) sin [( π s – ) × . s + π / ] = – ( . m)( π s – ) sin [( π + π / )] = π × . m s – = m s – Using Eq.( .
), the acceleration of the body