b’Subject: Physics’

b’Topic: Experiment F’

b’https://phet.colorado.edu/sims/html/masses-and-springs/latest/masses-and-springs_en.html\n\nPlease complete the following physics reponyzrt using the following link’

Title:

Experimentalist:

Experiment Date:

Abstract:

General Physics Lab (PHYS 2111) Exp. F, Hooke’s Law (Online Version) 2

1 Introduction: Hooke’s Law – Spring Force

1.1 Spring Constant, *k*

Hooke’s law explains the force by a spring, when it is strached or squeezed; The force by a spring is proportional

to the displacement !*x*, that is how far it is stracthed or squeezed;

*F*

spring = *!**k*!*x. *(1)

Here, *k *is the spring constant, which characterizes the sti↵ness of the spring.

When the weight is in equilibrium (*F ~*net = 0), the weight *W *= *mg*, which is the gravitational force by

Earth, is balanced out the spring force, *F*spring in Hooke’s law, Eq. (1). The magnitude of the hanging weight,

therefore, must be the same as the magnitude of the spring force. The spring constant of a spring can be

measured for a spring by using known wieghts.

*k *= *|**F*spring*|*

!*x *=

*W *!*x*

=

*mg*

!*x. *(2)

In this report, in order to explore Hooke’s law, the PhET simulation program developed by University

of Colorado Boulder[1], is used to get the measurements of springs. First, by measuring the displacement for

known mass, the spring constant, *k*, will be obtained. Then this value of the spring constant will be used to

decide the masses of unkown weights.

1.2 Harmonic Motion of Spring

In this experiment, we will explore the simple harmonic motion (oscillation) of a spring as well. By using

Newton’s second law with Hooke’s law, the period of the harmonic motoin can be

*T *= 2*⇡*r*m k , *(3)

where *m *is the mass of the object suspended to the spring. From the formula, we expect;

*• *Period of harmonic motion of a spring dese not depend on the amplitude.

*• *Period of harmonic motion of a spring only depends on the mass of the object attached.

In this experiment, we will check these two aspects of the simple harmonic motion of a spring. In addition,

by using the data taken with Eq. (3), we can estimate the spring constant, *k *of the spring;

*k *= 4*⇡*2*m*

*T *2 *. *(4)

2 Equipments – Device / Samples:

PhET Interactive Simulation of Hooke’s Law Equipment[1]

General Physics Lab (PHYS 2111) Exp. F, Hooke’s Law (Online Version) 3

3 Detail Procedures:

3.1 Spring Constant, *k*

a) By clicking the link on Canvas, open Hooke’s Law Equipment of the PhET Interactive Simulation. You

will see four items under the title ‘Masses and Springs’.

b) Choose Intro by clicking on it. On the display,

i. set ‘Spring Constant 1’ by moving the cyan block under. A smaller value might be better for the

experiment. Once you set the constant, then it must be fixed through the experiment. So be careful

not to change it by mistake. Also make to check ‘Spring Constant’ below in the box.

ii. On the box in the right top corner, check ‘Natural Length’ and ‘Equilibrium Position’. You will

have two dotted lines (blue and green) on the spring.

iii. Just below, make sure the gravity set to be ‘Earth’ and ‘Damping = 0’.

c) Find the weights on the left bottom corner. By using the mouse, pick up ‘50 g’ and hang it at the bottom

end of the left spring (Spring 1). Try to find the point which the weight is not bouncing. Measure

the displacement for the weight of ‘50 g’, the length between ‘Natural Length (blue dotted line)’ and

‘Equilibrium Position (green dotted line)’. by using the yellow ruler in the box in the middle of the

right hand side and record it in Table 1.

d) In the same way, measure displacements for ‘100 g’, and ‘250 g’ and add them to Table 1.

e) In Table 1, obtain spring constants for weights and the average value of the constant.

f) Similarly, measure displacements for unknown weights next to the known ones, and add them to Table 2.

g) In table 2, find the masses of unknown weights.

3.2 Harmonic Motion of Spring

a) Choose the weight of 100 g to suspend at the end of Spring 1.

b) Check the box for ‘Movable Line’ in the right top box, then red dotted line will show up. Using the ruler,

move the red line at 15 cm from the equilibrium (green line). Move the weight at the red line and release

it.

c) Get the stop watch from the tool box in the middle of the right hand side. By using the watch, measure

the time for 10 periods, six times in Table 3.

d) Measure the period of the harmonic motion with di↵erent amplitudes in Table 3. Compute the standard

deviation for the statistical uncertainty of period measurements in in Table 3.

e) By making the graph of the amplitude versus period in Figure 2, explore the independence of period on

amplitude.

f) Now fix the amplitude at 20 cm by moving the red line and ruler. Try five di↵ernt weights including two

of unknown weights. Again measure 10 periods (10*T*), five times each to fill in Table 4. Compute the

standard deviation for the statistical uncertainty of period measurements in in Table 4.

g) By making the graphs of the mass versus period in Figure 3 and the mass versus period squared in Figure

4, explore the mass dependence of period.

General Physics Lab (PHYS 2111) Exp. F, Hooke’s Law (Online Version) 4

4 Data and Data Analysis

4.1 Spring Constant, *k*

The displacements of ‘Spring 1’ are measured for three di↵erent weights hung in Table 1. Using Eq. (2), the

Mass Weight Displacement Spring Constant

*m*[g] *W *= *mg*[N] !*x*[cm] *k *= *F*spring !*x *= ! *W x*[N/cm]

50

100

250

Average *k *¯

Table 1: Measuring Displacements for Known Weights to measure Spring Constant of ‘Spring 1’

average spring constant of the spring is obtained in the same table.

*k *¯ = N*/*cm (5)

Three unknown weight are suspended and the displacements are measured in Table 2.

Displacement Weight [N] Mass

Weights !*x*[cm] *W *= *F*spring = *k*!*x m *= *W*

*g*

[g]

Magenta

Cyan

Orange

Table 2: Measuring Displacements to Measure Unknown Weights by using the Spring Constant obtained

Using the value of the spring constant *k *in Eq. (5), the masses of these unknow weights are measured in

the same table.1 These measured values in Table 2 will be used in the next section when the harmonic motion

of the spring is tested for the dependence of peiod on mass.

Figure 1 shows the relationship between the weight and displacement, Hooke’s law. In this graph, all six

weights, both the konwn and unknown weights, are used.

1The units need to be considered very carefully. Grams (g) is used for masses here, while newtons ([N = kg m/s2) is used for

the spring force. You would, therefore, need to convert (g to (kg, and back to (g for the masses of those unknow weights.

General Physics Lab (PHYS 2111) Exp. F, Hooke’s Law (Online Version) 5

Figure 1: Displacement !*x *[m] vs. Weight *W *[N]

4.2 Harmonic Motion of Spring

The period of harmonic motion of Spring 1 is measured with di↵erent amplitude in Table 3. The same

weight of 100 g was used for all amplitudes. The average and statistical uncertainty of each measurement is

also calculated in the same table. The graph in Figure 2 shows the dependence of period on the amplitude of

this harmonic motion.

The period of harmonic motion of Spring 1 is measured with various weights with the amplitude fixed

at 20 m in Table 4. Five di↵erent weights, including two unknowns (Cyan and Orange), are used in this

measurement. For those unknowns, the values of masses measured in Table 2 was used. The average and

statistical uncertainty of each measurement is also calculated in the same table. The graph in Figure 3 shows

the dependence of period on mass of this harmonic motion.

From Eq. (3), we have the linear relationship between mass *m *and period squarded *T *2;

*T *2 = 4*⇡*2

*k m *(6)

The graph of the mass (*m*) versus period squared (*T *2) is shown in Figure 4. The linear regression technique

is applied in Table 5 to get the proportional coe#cent of this linear relationship.

Slope : 4*⇡*2

*k *= *a *=

*N*⌃*x**i**y**i **! *⌃*x**i*⌃*y**i*

*N*⌃*x*2

*i **! *(⌃*x**i*)2 = [Unit : ]

*y **! *intercept : *b *= ⌃*x*2 *i *⌃*y**i **! *⌃*x**i*⌃*x**i**y**i*

*N*⌃*x*2

*i **! *(⌃*x**i*)2 = [Unit : ]

The spring constant *k *of the spring can be measured from the slope obtained here;

*k *= 4*⇡*2

*a*

= [Unit : ]*,*

General Physics Lab (PHYS 2111) Exp. F, Hooke’s Law (Online Version) 6

Amplitude *A*[cm] 10 Periods 10*T*[s] Period *T*[s] *T **! **T *¯ [s] (*T **! **T *¯)2 [s2]

15

Mean [s] Statistical Error [s]

*T *¯ = P*i **T**i*

*N **#**T *= qP*i**N *(*T**! ! *1*T *¯)2

30

Mean [s] Statistical Error [s]

45

Mean [s] Statistical Error [s]

60

Mean [s] Statistical Error [s]

Table 3: Period of Spring Harmonic Motion with Various Ampitudes with the weight of 100 g

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