The speed of a body of mass m, when it is at a distance x from the orign, is v. Its total energy is `1/2mv^(2)+1/2kx^(2)`, where k is constant. Prove that the body executes a simple harmonic motion.

Explanation

Total energy, `E=1/2mv^(2)+1/2kx^(2)`
`therefore" "(dE)/(dt)=1/2md/(dt)(v^(2))+1/2kd/(dx)(x^(2))`
= `1/2md/(dv)(v^(2))(dv)/(dt)+1/2kd/(dx)(x^(2))(dx)/(dt)`
= `1/2m*2v*a+1/2k*2x*v`
= mva + kxv [a = acceleration]
If E = constant, `(dE)/(dt)=0`.
`therefore" "mva+kxv=0or,a=-k/mx=-omega^(2)x[omega=pmsqrt(k/m)]`
This is the equation of a simple harmonic motion.