The function `x = A sin^2 (omega)t + B cos^2 (omega)t + Csin (omega)t cos (omega)t` represent (SHM) for which of the option(s).

Options

A. for all value of `A, B and C(C != 0)`.
B. A = B, C = 2 B
C. A = - B, C = 2 B
D. A = B, C = 0

Answer

A, B, C

Explanation

The given equation is
`x = A sin^2 omega t + B cos^2 omega T + C sin omega t cos omega t`
Rearranging the equation in a meaningful form (for interpretation of (SHM)
`s = (A)/(2) (2 sin^2 omega t) + (B)/(2) (2 cos^2 omega t)+ (C)/(2) (2 sin omega t cos omega t)`
=`(A)/(2) [1 - cos 2 omega t] + (B)/(2) [1 + cos 2 omega t]+ (C)/(2) [sin 2 omega t]`
(a) For `A = 0 and B = 0, x = (C)/(2) sin (2 omega t)`
The above equation is that of (SHM) with amplitude `(C)/(2)` and angular frequency `2 omega`. Thus option (a) is correct.
(b) If `A = B abd C = 2B then x = B + B sin 2 omega t`
This is equation of (SHM). The mean position of the particle executing (SHM) is not at the origin. option (b) is correct.
( c) `A = - B, C = 2 B ,` Therefore
`x = B cos 2 (omega) t = 2 B sin 2 (omega) t`
Let `B = X cos phi + X cos 2 (omega) t sin (phi)` ltbgt This represents equation of (SHM).
(d) `A = B, C = 0 and x = A`. This equation does not represents (SHM).