Energy in Simple Harmonic Motion
In Chapter 8 we saw that the energy of a linear oscillator transfers back and forth between kinetic ‘energy and potential energy, while the sum of the two-the mechanical energy E of the osciIIator-remains constant. We now consider this situation quantitatively. The potential energy of a linear osciIIator like that of Fig. 16-5 is associated entirely with the spring. Its value depends on how much the spring is stretched or compressed-that is, on .\'(1). We can use Eqs. 8-11 and 16-3 to find Note carefulty that a function written in the fonn coa2 ,4 (as here) meaIJS (COS,4)2
and is not the same asone written cos ,42, which means cos (,42). .
The kinetic energy of the system of Fig. 16-5 is usoc:iated entirely with the
block. Its value depends on how fast the block is moving-that is, on Y(t). We can
use Eq. 16-6 to find
K(t) = !mv2 = !m(.h~ sin2(wt + 4iJ.
If we use Eq. 16-12 to substitute k/m for w’-, we can write Eq. 16-19 as
K(t) = !mv2 = !h! sinl(~ + ~).
The mechanical energy follows from Eqs. Wi-II an4 16-20 ami it
= !kx~ cos2(wt + ~) + !kx~ sin2(~ + ~)
= !kx~[CQS2(fdt + ~) + sin2(1&It+ .)].
Fig. 16-6 (a) Potential energy V(/),
kinetic energy K(/), and mechanical .
energy E as functions of time I for a
linear harmonic oscillator. Note that all The mechanical energy of a linear oscillator is indeed constant and independent of
energies are positive and that the po- time. The potential energy and kinetic energy of a linear oscillator are shown as
tential energy and the kinetic energy functions of Vme t in Fig. 16-60, and as functions of displacement x in Fig. 16-6b.
peale twice during every period. (b) Po- You might now understand why an oscillating system nonnaJly contains an
tential energy V(x), kinetic energy K(x), element of springiness and an element of inertia: The former stores its potential
and mechanical energy E as func- energy and the latter stores its kinetic energy. tions of position x for a linear harmonic oscillator with am.”litude. x. ~or ~CHECKPOINT 3: In Fig. 16-5, the block has a kinetic energy Qf 3 J and the spring has x = 0 the .e~ergy IS all ~tnetlc, and for an elastic potential energy of 2 J when the block is at x = +2.0 em. (a) What is the kix
= :tx”, 11 IS all potential. netic energy when the block is at x = O? What are the elastic potential energies when the block is at (b) x = -2.0 em and (c) x = -x”, V(,,) For any angle a. ‘Thus, the quantity in the square brackets a(16-18.