Name: _____________________________________Date: ___/____/____ Course/Lab Section: _______
Name of Lab Instructor: _________________________
PreLab
The Simple Pendulum
Instructions: Prepare for this lab activity by answering the questions below. Note that this is a PreLab. It
must be turned in at the start of the lab period. Time cannot be given during the lab activity for PreLab work.
After the start of lab activities PreLabs cannot be accepted.
Q1. What is the driving force behind the back-and-forth oscillatory motion of the simple pendulum?
Q2. Why pendulum clocks need springs or weights besides the pendulum?
Q3. A pendulum clock would not work well on board of a ship in the ocean. Why?
Q4. On a fixed location, if you want to double the period of a pendulum, by how much do you need to change
its length?
Q5. Consider a pendulum with a period of 1.0 s on Earth. What would be the period of this pendulum on the
Moon were the acceleration of gravity is about 1/6 the value of the acceleration of gravity on Earth?
Q6. Explain the small angle approximation and give a numerical example to illustrate it.
The Simple Pendulum
Purpose: The purpose of this lab activity is to measure the local acceleration of gravity g using the equation
for the simple pendulum
T  2
(1)
g
where T is the period of the pendulum, is its length, and g is the acceleration of gravity.
This lab activity is intended to assist you to:
 Develop a better understanding of the relationship between the period T and the length of the simple

pendulum.

 Perform experimental measurements
that allow the determination of the local acceleration of gravity g.
Introduction: The simple pendulum consists of a massive ball suspended by 
a light flexible string. The
length of the pendulum is measured from a fixed support to the center of mass of the ball. The period is the
time for one complete oscillation. Equation (1) above is applicable only for oscillations with small amplitude
because it is derived using the so-called small angle approximation. The last part of this lab activity involves
deriving Eq. (1) and showing that the approximation is valid. Therefore we need to keep the angles of
 oscillation small as we proceed through the experiment. Assuming g to be constant at any given location,
though it varies from place to place, we are left with two variables in Eq. (1): the period T and the length .
We will vary the length of the pendulum and measure the corresponding period T for a number of values
of .

Procedure:

Part I: Verification
of Eq. (1)

a. Cut a piece of string long enough to reach from the top of the pendulum support to the floor. Run the
string through the hole in the ball and tie a knot in it. Place the pendulum clamp at the top of the ring stand
and clamp the ring stand to the table such that, when the pendulum is suspended it will be out beyond the
edge of the table but as near the vertical support as possible. This is to allow for the pendulum to be longer
than the ring stand.
b. On the table, place the ball at some convenient point on the double meter stick such as the 10 cm mark.
The center of mass of the ball should be at this mark. Stretch the string out along the meter stick and grip it
with a thumb nail 10 cm from the center of the ball. Pass the string up through the pendulum clamp until the
thumb nail is against the support. Clamp the string in this position and check the length of the pendulum. It
should be 10 cm long as measured from the center of the ball to the bottom of the support.
c. Displace the ball to one side slightly and release it. The angle the string makes with the vertical should be
no more than 5. Using the stop watch time 50 complete oscillations. The average period is the total time
divided by 50. Record the period on the DATA TABLE.
d. Repeat steps b and c for each of the lengths listed on the DATA TABLE.
Analysis I
Plot a graph of versus T to as a large a scale as convenient using the values from the DATA TABLE. Label
it Graph I and use it to determine: (i) the relation between and T , and (ii) the periods of a 5.0 cm and of a
3.0 m long pendulum. Record your results on the DATA TABLE.
Analysis II
 all values of T and record them on the DATA TABLE. Plot a graph of versus T 2 to as large a scale
Square

as convenient and label it Graph II. Write the equation
for the line letting k represent the slope and use it to
determine the periods of a 5.0 cm and of a 3.0 m long pendulum. Record your results on the DATA TABLE.
Analysis III
 the constant is the same constant k
Solve Eq. (1) for . You should get equals a constant times T 2, where
representing the slope in Analysis II above. Use this fact to obtain a numerical value for g. Compare your
result with the known value of 9.80 m/s2, calculate and record your percentage error on the DATA TABLE.


Part II: Validity of the small angle approximation in Eq. (1)
The right side of the figure below represents a pendulum swung to the left side of its equilibrium position.
The angle  shown here is much larger than the angle that should be used in the experiment. We use a larger
angle to facilitate visualization and analysis. The left side of the figure is the corresponding force
Illustration of the simple pendulum and the corresponding force diagram.
diagram. It displays the weight W = mg and its components along the direction of motion Wa, and along the
perpendicular direction Wl (equal in magnitude to the tension on the string and opposite in direction to it).
Wl and the tension on the string make no contribution to the motion. However, the other component Wa is
unbalanced and generates an acceleration a according to Newton’s second law:
Wa = m a
(2)
Notice from the figure that
Wa = W sin
(3)
where W = mg and sin  
s
. Therefore,
ag
s
(4)
Notice from the figure that x is the displacement of the pendulum while s is the corresponding horizontal
 The arc x is the length of the curved trajectory of the pendulum. The small angle approximation
component.
assumes that these three quantities are approximately equal when the angle  is sufficiently small. In this
case, the angle  in radians (which by 
definition is the ratio between the arc length and the radius of the
trajectory) is approximately equal to sin .
Analysis I
Compute the angle  = 5 in radians, and sin also. Record these values on DATA TABLE and determine
the corresponding percentage difference. Repeat for  = 15 and  = 30.
Analysis II
From the definition of simple harmonic motion, it can be shown that the acceleration for any object
executing such a motion is given by
(5)
a  4  2 f 2x
where f represents the frequency of the oscillatory motion and is given by f = 1/T .
On the back of the DATA TABLE sheet use Eq. (4) with s replaced by x, and Eq. (5) to derive Eq. (1).

The Simple Pendulum
DATA TABLE
Name (Partner A): __________________________________________
Date: ____________
Name (Partner B): __________________________________________
Course/Lab Section: __________
Name (Lab Instructor): ______________________________________
(cm)
(m)
T2
(s2)
T
(s)
10

20
30
40
50
60
80
100
120
140
160
From Analysis I:
When = 5.0 cm, T = __________. When
= 3.0 m, T = __________.

From Analysis II:
When = 5.0 cm, T = __________.
 When
= 3.0 m, T = __________.

From Analysis III:
g = _____________.
% Error
= _________
From Analysis IV:
5 = ___________rad
sin5 = ____________
% Difference:____________
15 = ___________rad
sin15 = ____________
% Difference:____________
30 = ___________rad
sin30 = ____________
% Difference:____________
SHOW ALL YOUR WORK.