Question

A body of mass m performs linear S.H.M. given by equation, x = P sin cot + Q sin`(omega"t" + pi/2)`. The total energy of the particle at any instant is ______.

विकल्प

• `1/2 "m"omega ("P"^2 + "Q"^2)`

• `1/2 "m"omega^2//"PQ"`

• `1/2 "m"omega^2 "PQ"`

• `1/2 "m"omega^2 "P"^2 "Q"^2`

Answer 1

A body of mass m performs linear S.H.M. given by equation, x = P sin cot + Q sin`(omega"t" + pi/2)`. The total energy of the particle at any instant is `underline(1/2 "m"omega ("P"^2 + "Q"^2))`.

Explanation:

x = P sin cot + Q sin`(omega"t" + pi/2)`

= P sin ωt + Q cos ωt

Let P = A cos θ    Q = A sin θ

∴ P² + Q² = A²

∴ x = A sin (ωt + θ)

`"E"_"total" = 1/2 "m"omega^2 "A"^2`

`= 1/2 "m"omega^2 ("P"^2 + "Q"^2)`