If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, show that the expressions (aT)/xanda^(2)T^(2)+4pi^(2)v^(2) do not change with time.

If x, v and a denote the displacement, the velocity and the accelerati

1.	If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, show that the expressions (aT)/xanda^(2)T^(2)+4pi^(2)v^(2) do not change with time.
Answer» Solution :If A is the AMPLITUDE, `omega` is the angular frequency and x is the displacement of a particle executing SHM then, velocity `v=pmomegasqrt(A^(2)-x^(2))` acceleration `a=-omega^(2)x` time PERIOD `T=(2PI)/omega` `(aT)/x=(-omega^(2)x*(2pi)/omega)/x=-2piomega` = constant `a^(2)T^(2)+4pi^(2)v^(2)=(-omega^(2)x)^(2)((2pi)/omega)^(2)+4pi^(2)omega^(2)(A^(2)-x^(2))` `=omega^(4)x^(2)(4pi^(2))/omega^(2)+4pi^(2)omega^(2)A^(2)-4pi^(2)omega^(2)x^(2)` `=4pi^(2)omega^(2)A^(2)` = constant