A particle is executing SHM along a straight line. Its velocities at distances X1 and X2 from mean position are V1 and V2, respecilvely. Its time period is :

A
$large 2pi sqrt {frac{{V_1^2 + V_2^2}}{{X_1^2 + X_2^2}}}$
B
$large 2pi sqrt {frac{{V_1^2 - V_2^2}}{{X_1^2 - X_2^2}}}$
C
$large 2pi sqrt {frac{{X_1^2 + X_2^2}}{{V_1^2 + V_2^2}}}$
D
$large 2pi sqrt {frac{{X_1^2 - X_2^2}}{{V_1^2 - V_2^2}}}$

Solution:

$large {v_1}, = ,omega sqrt {{A^2} - x_1^2}$

$large Rightarrow ,frac{{V_1^2}}{{{omega ^2}}}, = ,{A^2} - x_1^2, Rightarrow ,{A^2}, = ,frac{{V_1^2}}{{{omega ^2}}} + x_1^2$

$large {V_2}, = ,omega sqrt {{A^2} - x_2^2}$

$large Rightarrow ,frac{{V_2^2}}{{{omega ^2}}}, = ,{A^2} - x_2^2, Rightarrow ,{A^2} = ,frac{{V_2^2}}{{{omega ^2}}} + x_2^2$

$large frac{{V_1^2}}{{{omega ^2}}} + x_1^2, = ,frac{{V_2^2}}{{{omega ^2}}} + x_2^2, Rightarrow ,frac{{V_1^2 - V_2^2}}{{{omega ^2}}}, = ,x_2^2 - x_1^2$

$large {omega ^2}, = frac{{V_1^2 - V_2^2}}{{x_2^2 - x_1^2}}, Rightarrow ,T, = ,2pi sqrt {frac{{x_2^2 - x_1^2}}{{V_1^2 - V_2^2}}}$

A particle is executing SHM along a straight line. Its velocities at distances X₁ and X₂ from mean position are V₁ and V₂, respecilvely. Its time period is :

Solution:

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