An SHM is given by the equation ` x = 8 sin (4 pi t) + 6 cos (4pit)]` cm find its amplitude and period

Explanation

Data x =`[ 8 sin (4pit) + 6 cos (4pit)]` cm
` x = 8 sin (4 pit)+ 6 cos (4 pit)`
` 8 sin (4 pit) + 6 sin (4pit + pi/2)`
Thus, x is the superposition of two parallel SHMs of the same period : ,
` x = x_(1) + x_(2) ` where
` x_(1) = 8 sin (4pit) cm= A_(1) sin (omegat + prop)`and
` x_(2) = 6 sin (4pit + pi/2) cm = A_(2) sin (omegat + beta)`
`A_(1)= 8 cm , A_(2) = 6 cm, omega = 4 pi " rad/s" , prop=0 , beta = pi/2` rad
(1)Resultant amplitude
` |R|= sqrt(A_(1)^(2)+2A(1)A_(2)cos(prop-beta))`
` = sqrt((8)^(2) + (6)^(2) + 2 xx 8 xx 6 cos (0 -pi//2))`
` sqrt(100) = 10 ` cm
(2) Period , ` T= (2pi)/(omega) = (2pi)/(4 pi) = 1/2` s