EXPERIMENT #1SIMPLE PENDULUM AND MASS-SPRING

SYSTEM IN SIMPLE HARMONIC MOTION

Proponents:

Amaya, Marife

Batulan, Krizella

Infante, Diane

Jugalbot, Lydel

Objectives:

To validate the equation 𝑇 = 2𝜋𝐿

𝑔using 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

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.

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

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 = 2𝜋𝐿

𝑔, where g is the

acceleration due to gravity.

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.

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 = 2𝜋𝑚

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,

𝒎𝒆𝒇𝒇 = 𝒎𝒔𝒖𝒔𝒑𝒆𝒏𝒅𝒆𝒅 𝒐𝒃𝒋𝒆𝒄𝒕 +𝟏

𝟑𝒎𝒔𝒑𝒓𝒊𝒏𝒈

Materials:

Helical spring

Set of weights

Meter stick

Stand or supporting rod

Stopwatch

String

Pendulum bob

Protractor

Digital balance

Mass hanger or hook

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.

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.

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 .

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.

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.

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.

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.