Question

A body performing a simple harmonic motion has potential energy 'P₁' at displacement 'x₁' Its potential energy is 'P₂' at displacement 'x₂'. The potential energy 'P' at displacement (x₁ + x₂) is ________.

विकल्प

• P₁ + P₂

• `sqrt("P"_1^2 + "P"_2^2)`

• `"P"_1 + "P"_2 + 2sqrt("P"_1 "P"_2)`

• `sqrt("P"_1 "P"_2)`

Answer 1

A body performing a simple hannonic motion has potential energy 'P₁' at displacement 'x₁' Its potential energy is 'P₂' at displacement 'x₂'. The potential energy 'P' at displacement (x₁ + x₂) is `underline("P"_1 + "P"_2 + 2sqrt("P"_1 "P"_2))`.

Explanation:

`"P"_1 = 1/2"kx"_1^2`

`Rightarrow "x"_1^2 =(2"P"_1)/"k"`

`"P"_2 = 1/2"kx"_2^2`

`Rightarrow "x"_2^2 = (2"P"_2)/"k"`

`"P" = 1/2"k" ("x"_1 + "x"_2)^2`

`= 1/2"k" ("x"_1^2 + "x"_2^2 + 2"x"_1 "x"_2)`

`= 1/2"k" ((2"P"_1)/"k" + (2"P"_2)/"k" + 2 sqrt ((2"P"_1)/"k". (2"P"_2)/"k"))`

`"P" = 1/2"k" xx 2/"k" ("P"_1 + "P"_2 + 2 sqrt("P"_1 "P"_2))`

`"P" = "P"_1 + "P"_2 + 2 sqrt("P"_1 "P"_2)`