Question

A particle is performing harmonic motion if its velocity are `v_(1)` and `v_(2)` at the displecement from the mean position are `y_(1)` and `y_(2)` respectively then its time period is
A. `2 pi sqrt((x_(2)^(2) - x_(1)^(2))/(V_(1)^(2) - V_(2)^(2)))`
B. `2 pi sqrt((V_(1)^(2) + V_(2)^(2))/(x_(2)^(2) + x_(1)^(2)))`
C. `2 pi sqrt((V_(1)^(2) - V_(2)^(2))/(x_(2)^(2) - x_(1)^(2)))`
D. `2 pi sqrt((x_(1)^(2) - x_(2)^(2))/(V_(1)^(2) - V_(2)^(2)))`

Answer 1

`v = omega sqrt(a^(2) - x^(2)) implies v^(2) = omega^(2) (a^(2) - x^(2))`
`V_(1)^(2) = omega^(2) (a^(2) - x_(2)^(2)) = omega^(2) a^(2) - omega^(2) x_(2)^(2)`
`V_(1)^(2) - V_(2)^(2) = omega^(2) (x_(2)^(2) - x_(1)^(2))`
`omega = sqrt((V_(1)^(2) - V_(2)^(2))/(x_(2)^(2) - x_(1)^(2))) implies (2 pi)/(T) = sqrt((V_(1)^(2) - V_(2)^(2))/(x_(2)^(2) - x_(1)^(2))`