University Physics I - Classical Mechanics, 2019

262 CHAPTER 11. SIMPLE HARMONIC MOTION
external force is constant, and does not change direction, this work will be positive half the time,
and negative half the time. If it is kinetic friction, then of course it will change direction every half
cycle, and the work will be negative all the time.
In the case shown in Figure 11.6, the external force is gravity, which we know to be a conservative
force, so the energy that will be conserved will be the total energy of the system that includes the
oscillation and the Earth, and hence also the gravitational potential energy (for which we can use
here the familiar form U G = mgy):
E osc+earth = U spr + K + U G = 1 2 (y − y 0) 2 + 1 2 mv2 + mgy = const (11.17)
The reason it is no longer possible to combine the terms U spr + K into the constant 1 2 kA2 ,asin
Eq. (11.14), is that now we have
y(t) =y 0 ′ + A cos(ωt + φ)
v(t) =−ωA sin(ωt + φ) (11.18)
so the oscillations are centered around the new equilibrium position y 0 ′ , but the spring energy is
not zero at that point: it is zero at y = y 0 instead. You can check for yourself, however, that
if you substitute Eqs. (11.18) intoEq.(11.17), and make use of the fact that k(y 0 ′ − y 0)=−mg
(Eq. (11.15)), you do indeed get a constant, as you should.
11.3 Pendulums
11.3.1 The simple pendulum
Besides masses on springs, pendulums are another example of a system that will exhibit simple
harmonic motion, at least approximately, as long as the amplitude of the oscillations is small. The
simple pendulum is just a mass (or “bob”), approximated here as a point particle, suspended from
a massless, inextensible string, as in Fig. 11.7 on the next page.
We could analyze the motion of the bob by using the general methods introduced in Chapter 8 to
deal with motion in two dimensions—breaking down all the forces into components and applying
⃗F net = m⃗a along two orthogonal directions—but this turns out to be complicated by the fact
that both the direction of motion and the direction of the acceleration are constantly changing.
Although, under the assumption of small oscillations, it turns out that simply using the vertical
and horizontal directions is good enough, this is not immediately obvious, and arguably it is not
the most pedagogical way to proceed.