Solved Example Problems for Energy in Simple Harmonic Motion

EXAMPLE 10.15

Write down the kinetic energy and total energy expressions in terms of linear momentum, For one-dimensional case.

Solution

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

EXAMPLE 10.16

Compute the position of an oscillating particle when its kinetic energy and potential energy are equal.

Solution

Since the kinetic energy and potential energy of the oscillating particle are equal,

1/2 mω 2 (A2 âˆ’ x 2 ) = 1/2 mω2 x2

A2 âˆ’ x2 = x2

2x2 = A2

x = ±A/√2