1) The experiment aims to validate equations for the period of a simple pendulum and a mass-spring system undergoing simple harmonic motion. 2) A simple pendulum and mass-spring system are used to determine the acceleration due to gravity and spring constant. 3) Variables like length, mass, angle of displacement are changed to study their effect on the period of oscillation based on theoretical equations.

1. EXPERIMENT #1
SIMPLE PENDULUM AND MASS-SPRING
SYSTEM IN SIMPLE HARMONIC MOTION
Proponents:
Amaya, Marife
Batulan, Krizella
Infante, Diane
Jugalbot, Lydel

2. Objectives:
 To validate the equation 𝑇 = 2𝜋
𝐿
𝑔
using a constant mass
of the bob and an angular displacement of less than 10°.
 To determine the acceleration due to gravity , g, using the
concept of SHM
 To understand the behavior of objects in simple harmonic
motion by determining the spring constant of a mass-
spring system
 To determine the spring constant, k, using Hooke’s law
and Simple Harmonic Motion

3. Theory:
A particle that vibrates vertically in simple harmonic motion
moves up and down between two extremes y = ±A. The maximum
displacement A is called the amplitude. This motion is shown
graphically in the position-versus-time plot in Fig. 1.

4. One complete oscillation or cycle or vibration is the motion from, for
example, y = −A to y = +A and back again to y = −A. The time interval T required
to complete one oscillation is called the period.
An example of simple harmonic motion that we will investigate is
the simple pendulum. The simple pendulum consists of a mass m, called the
pendulum bob, attached to the end of a string. The length L of the simple
pendulum is measured from the point of suspension of the string to the center of
the bob as shown in Fig. 2 below.
Fig. 2. The Simple Pendulum Fig. 3. Components of the forces

5. If the bob is moved away from the rest position through some
angle of displacement θ as in Fig. 2, the restoring force will return the
bob back to the equilibrium position. The forces acting on the bob are
the force of gravity and the tension force of the string. The tension
force of the string is balanced by the component of the gravitational
force that is in line with the string (i.e. perpendicular to the motion of
the bob). The restoring force here is the tangential component of the
gravitational force as shown in Fig. 3 above.
In accordance with Newton’s second law of motion, the
period of oscillation of a simple pendulum for small angular
displacements ( < 10° ) can be derived as : 𝑇 = 2𝜋
𝐿
𝑔
, where g is the
acceleration due to gravity.

6. Another example of SHM is a mass spring system. A mass
suspended at the end of a spring will stretch the spring by some distance y. The
force with which the spring pulls upward on the mass is given by Hooke’s Law
𝐹 = −𝑘𝑦, where k is the spring constant and y is the stretch in the spring when
a force F is applied to the spring. The spring constant k is a measure of the
stiffness of the spring. The spring constant can be determined experimentally by
allowing the mass to hang motionless on the spring and then adding additional
mass and recording the additional spring stretch as shown below.

7. When the mass is motionless, its acceleration is zero. According to
Newton's second law the net force must therefore be zero. There are two
forces acting on the mass; the downward gravitational force and the upward
spring force. See the free-body diagram above.
Applying Newton’s second law, the period of oscillation for a mass-spring
system is given in the equation 𝑇 = 2𝜋
𝑚
𝑘
, where m is the mass of the body
and k is the force constant of the spring.
However, the suspended mass is not the only moving mass because the
spring is in motion as well. Thus, m in the above equation is the effective mass
which is a combination of the mass of the suspended object and a part of the
mass of the spring. It is found by analysis that one-third of the mass of the
spring must be added to the mass of the suspended object, thus,
𝒎 𝒆𝒇𝒇 = 𝒎 𝒔𝒖𝒔𝒑𝒆𝒏𝒅𝒆𝒅 𝒐𝒃𝒋𝒆𝒄𝒕 +
𝟏
𝟑
𝒎 𝒔𝒑𝒓𝒊𝒏𝒈

8. Materials:
 Helical spring
 Set of weights
 Meter stick
 Stand or supporting rod
 Stopwatch
 String
 Pendulum bob
 Protractor
 Digital balance
 Mass hanger or hook

9. Procedure:
A. Determination of the acceleration due to gravity
Parameter: LENGTH
Assemble the simple pendulum set- up. Adjust the length of
the pendulum to 30 cm, this will serve as our first of the five varying
lengths that would be experimented. With a meter stick, measure the
length from the point of support to the center of the bob. Then displace
the bob to one side with an angle of not more than 100. After releasing
the bob make sure it is moving freely and not in a circular manner but
back and forth. Start the stopwatch when the bob is at one end and
record in Table 1, the time it takes the pendulum to make 20 complete
oscillations. Make 5 trials. Then the period of the pendulum could be
calculated from the measurements taken. Repeat this step for lengths
of 40 cm, 50 cm, 60 cm, 70 cm and 80 cm.

10. Procedure:
A. Determination of the acceleration due to gravity
Parameter: MASS
Using the same set-up, adjust the length of the pendulum to 40 cm. This will
serve as the constant length of the pendulum as the weight added varies. A
weight hanger and a set of varying weights are used instead of a bob.
Suspend a 30-gram slotted mass together with the weight hanger of
predetermined mass on the string of the pendulum. Then displace the first
weight to one side with an angle of not more than 100. After releasing the
weight hanger, make sure it is moving freely and not in a circular manner but
back and forth. Start the stopwatch when the weight is at one end and record
in Table 2, the time it takes to make 20 complete oscillations. Make 5 trials.
Then the period could be calculated from the measurements taken.
Repeat this step with an addition of a 20-gram load in order for the next
weight to be 50 grams. Continue adding 20-gram load only until it reaches
110 grams (excluding the mass of the weight hanger) having five varying
masses namely: 30-g, 50-g, 70-g, 90-g and 110-g.

11. Procedure:
A. Determination of the acceleration due to gravity
Parameter: Angle Displacement
Use the same set-up, with the length of the pendulum still at 40 cm.
This will serve as the constant length of the pendulum as the angle
displacement of the bob varies. First, displace the bob to one side with
an angle of 50. After releasing the bob make sure it is moving freely and
not in a circular manner but back and forth. Start the stopwatch when
the bob is at one end and record in Table 3, the time it takes the
pendulum to make 20 complete oscillations. Make 5 trials. Then the
period of the pendulum could be calculated from the measurements
taken.
Repeat this step for angle displacements of 100 ,150 ,200 and 250 .

12. Procedure:
A. Determination of the acceleration due to gravity
FOR EACH OF THE PARAMETERS:
Calculate for the square of the period T2 for each length and make a
graph of T2 along the y-axis while length, l along the x-axis and plot in
Excel. Use the trend line option in Excel to determine the slope of the
graph then determine the acceleration due to gravity from the obtained
slope. Calculate the percent difference of this value of g to its
theoretical value of 9.8 m/s2.

13. B. Determination of the Spring Constant
I. Using Hooke’s Law or Displacement Method
Measure the mass of the spring using the digital balance, then
construct the mass-spring system. Record the position of the lower end of
the spring by means of a meter stick placed vertically alongside the
suspended spring, this will be the zero-load reading. Add 50 g of masses
to the hanger and measure the elongation of the spring. Add masses in
steps of 20 g until 230 g to the hanger and for each additional mass,
measure the corresponding elongation y of the spring produced by the
weight of these added masses and then record the data. Make a graph of
m vs. y and plot in Excel, use the trend line option to determine the slope
of the graph. Determine the spring constant from the obtained slope.

14. II. Using Simple Harmonic Motion
Start with adding 50 g to the hanger. Pull the mass down a short
distance and let go to produce a steady up and down motion without side-
sway or twist. As the mass moves downward past the equilibrium point,
start the clock and count "zero." Then count every time the mass moves
downward past the equilibrium point, and on the 20th passage stop the
clock. Repeat this whole process two more times and determine an
average time for 20 oscillations. Determine the period from this average
value. Repeat what is done with the first 50g for the same masses used in
BI. Get the square of the period for every pendulum mass. Make a graph
of T2 vs. m and plot in Excel, use the trend line option to determine the
slope of the graph and calculate the spring constant from the slope
obtained. Compare the results of k obtained using Hooke’s Law and that of
using Simple Harmonic Motion.

15. III. Determination if the Calculation is Within the Margin of Error
Given the average value of the spring constants obtained,
measure the period using the equation , where m is the
masses used in the previous experiment. After solving for the period for
every given mass, get the mean of the periods calculated. From the mean,
solve for the standard deviation. Identify the 95% confidence interval using
the equation where is the computed mean, t is the given
value from the t-table which is 2.262, and s which is the computed
standard deviation, and n is the number of trials, which is 10. Having the
confidence interval, we can get the margin of error for the period. Check
the measured periods from procedure B if they are inside the margin of
error.