Write down the kinetic energy and total energy expressions in terms of linear momentum, For one-dimensional case.
Kinetic energy is KE= 1/2 mvx2
Multiply numerator and denominator by m
KE= [1/2m] m2 vx2 = [1/2m] (mvx )2 = [1/2m] px2
where, px is the linear momentum of the particle executing simple harmonic motion.
Total energy can be written as sum of kinetic energy and potential energy, therefore, from equation (10.73) and also from equation (10.75), we get
E= KE +U( x) = [1/2m] px2 + 1/2 mω2 x2 = constant
Compute the position of an oscillating particle when its kinetic energy and potential energy are equal.
Since the kinetic energy and potential energy of the oscillating particle are equal,
1/2 mω 2 (A2 − x 2 ) = 1/2 mω2 x2
A2 − x2 = x2
2x2 = A2
x = ±A/√2