The document analyzes whether a simple pendulum exhibits simple harmonic motion. It describes a simple pendulum as a mass attached to an inextensible string with negligible mass, suspended from a rigid support. The document then analyzes the forces on the pendulum mass using a radial and tangential coordinate system. It determines that the restoring force is proportional to the sine of the angular displacement, so a simple pendulum is not strictly in simple harmonic motion. However, for small angular displacements, the sine can be approximated as the angle, making the restoring force directly proportional to displacement and allowing the simple pendulum to be considered an example of simple harmonic motion.

1. The Simple Pendulum
Victoria Nguyen Phys 101 LO2

2. A simple pendulum consists of:
-a mass attached to an inextensible string of
negligible mass
-other end of string tied to a rigid frictionless
support

3. When released,
pendulum swings back
and forth by gravity,
which is known as the
restoring force because
it causes the object to
return to its equilibrium
position
Point of maximum
displacement (s)
Inextensible rope
s

4. Question:
Is a simple pendulum in simple
harmonic motion?
Let’s find out!

5. Analyzing the simple pendulum
 This picture shows a mass
hanging from a string of length L,
making an angle (θ) with respect
to the equilibrium position (θ = 0 )
 The displacement, s, along the
arc of the circle is related to the
angular displacement as
 s = Lθ

6. Analyzing the simple pendulum
Now let’s analyze the forces on the mass!
As we know, there is tension force
(T) exerted by the rope
And we also know that Fg = mg =
weight force of the mass
But what is Fgcosθ and Fgsinθ ?

7. Analyzing the simple pendulum
Introducing: Radial axis and Tangential Axis
To analyze pendulum’s motion,
we will use a coordinate system
with a radial axis and a tangential
axis.
Radial axis is along the length of
the string while the
Tangential axis is tangent to
circular motion of the mass
Both axis are perpendicular to one
another and their directions
change as mass oscillates
Radial axis
Tangential axis

8. Radial axis
 Since Fg makes an angle θ with the radial axis,
the component of weight along the radial axis
is Fg= mgcos θ
 Also, it points AWAY from the suspension point
while the tension force points TOWARD the
suspension point along the same radial axis
 Since there is no motion on the radial axis,
the radial components of the Fg and Ft are
equal!
 T - mgcos θ = 0
T = mgcos θ
Radial axis

9. Tangential Axis
 The tangential axis is tangent to the circular
motion of the mass
 It also provides the restoring force that pulls
the mass towards the equilibrium position
 Therefore, tangential component of weight
along the tangential axis is:
Fg = mgsin θ
Tangential Axis

10. So is the simple pendulum in simple
harmonic motion?
 RECALL: In simple harmonic motion, the force that oscillates the mass is
directly proportional to the displacement
 In the simple pendulum, the force that’s propelling the mass from swinging
left to right is: Fg = mgsin θ
 Since this force is directly proportional to sine of the angular displacement
(sin θ), the simple pendulum is NOT in simple harmonic motion!
 HOWEVER, there is an exception!

11.  The simple pendulum can be in simple harmonic motion if the angle of
displacement is very small!
 We can then use the small-angle approximation equation for the sine
function, which basically tells us that when the angle of displacement (sinx)
is very small, it is equal to the displacement (x)
 sinx = x
As you can see, the two graphs are so close to each other
that they are generally said to be equal to one another
when the angle displacement (x rad) is very small (0-0.26)
Figure 13-23, pg 358 in textbook
Physics for Scientists and Engineers

12. Conclusion
 Generally, the simple pendulum is not in simple harmonic motion but if its
angle displacement is small enough, using small-angle approximation, it will
be said to be in simple harmonic motion