Question

The velocities of a particle in SHM at positions x₁ and x₂ are v₁ and v₂ respectively, its time period will be ______.

विकल्प

• 2π `sqrt(((v_1^2-v_2^2))/((x_2^2-x_1^2)))`

• 2π `sqrt(((x_1^2+x_2^2))/((v_2^2-v_1^2)))`

• 2π `sqrt(((x_1^2-x_2^2))/((v_2^2-v_1^2)))`

• 2π `sqrt(((x_1^2+x_2^2))/((v_2^2+v_1^2)))`

Answer 1

The velocities of a particle in SHM at positions x₁ and x₂ are v₁ and v₂ respectively, its time period will be 2π `underlinebb(sqrt(((x_1^2-x_2^2))/((v_2^2-v_1^2))))`.

Explanation:

`"v"_1^2 = omega^2("A"^2-x_1^2)`    ...(i)

`"v"_2^2 = omega^2("A"^2-x_2^2)`    ...(ii)

Substituting equation (i) from equation (ii),

`"v"_2^2 = "v"_1^2 + omega^2(x_1^2-x_2^2)`

⇒ `"v"_2^2 - "v"_1^2 = omega^2(x_1^2-x_2^2)`

⇒ ω² = `("v"_2^2 - "v"_1^2)/(x_1^2-x_2^2)`

⇒ `((2pi)/"T")^2 = ("v"_2^2 - "v"_1^2)/(x_1^2-x_2^2)`

⇒ ` T^2/(4pi^2) = (x_1^2-x_2^2)/("v"_2^2 - "v"_1^2)`

⇒ T = 2π `sqrt((x_1^2-x_2^2)/(v_2^2-v_1^2))`