WWWwwwGraphical Analysis:

Force Vs Stretch of a Cylindrical Spring

Force Vs Stretch of a Tapered Spring

Force Vs Stretch of a Tapered spring (2)

Force Vs Stretch of Springs in Parallel

Force Vs Stretch of Springs in Series

Force Vs Stretch of a Rubber Band

Force Vs Square Root of Rubber Band Stretch

Force Vs Stretch of a Rubber Band

Procedure: A cylindrical spring will be hanged from a hook’s law apparatus. The hanger

contains a pointer that will be pointed to 0 on the ruler, this will be the initial position. The mass

of the hanger does not matter because the stretch measurement will start from where the hanger

is originally. Next, place the slotted weights on the hanger in .040kg increments until the limit of

the ruler is reached. Then, the masses will be converted to weight by multiplying by 9.8 m/s^2,

this is the force used to stretch the spring from position 1 to position 2 (x1-x2). Also the

stretched amounts will be read in centimeters but will be converted to meters. Then, the graph of

force vs. stretch will be plotted. From the graph, the slope will be obtained which is the value of

the spring constant K and its regression value R^2.

In part 2 of the experiment, a tapered spring will be set up in a table clamp with the vertical rod

and horizontal support clamp. Suspend the tapered spring with the narrow end up from the

horizontal support. Then, place the hanger so that this will correspond to x=o. The mass of the

spring is irrelevant again. Now, place the slotted weights on the hanger in .050kg increments

until the stretch amount is close to the original length of the unstreched spring. Also, find the plot

the force vs. stretch, find the slope and the precision value. Then, place two springs in parallel

and follow the previous procedure. Compare the results. After this, put two springs in series,

record values, follow the same procedures and compare values.

In the 3rd part of the experiment, place a rubber band in the set up used for part two. Hang 1/2Kg

on the rubber band. Place the 2 meter ruler next to the mass and place its indicator right at the

bottom of the 1/2 kg, this is x=0. Now, add 1/2kg increments up to 5kg. Repeat previous

procedure and plot the relationship between the force and the stretched amount. Calculate the

percent error. Then integrate from x=0 to x=.1 the first graph from part 1, one of the tapered

spring from part 2 and the rubber band graph also. Answer the following question: What is the

final velocity of the spring when returns to x=0 when 100 g are hanged with a displacement of 10

cm after it has been released. The Area under the graph is the potential energy stored on the

spring, the derivative is the spring constant.

Data Analysis

Part 1: Cylindrical Spring

m(kg) F=mg (N) X (cm) X (m)

.040 .392 2.9 .029

.080 .784 5.5 .055

.120 1.176 8.1 .081

.160 1.568 10.9 .109

.200 1.96

.240 2.352

.280 2.744

Part 2: Tapered Spring 1

m(kg) F=mg (N) X (cm) X (m)

.050 .49 5.20 .0520

.100 .98 10.1 .101

.150 1.47 15.2 .152

.20 1.96 20.2 .202

.25 2.45 24.7 .247

.30 2.94 29.80 .2980

.35 3.43 34.90 .3490

.40 3.92 40.0 .400

Tapered Spring 2

m(kg) F=mg (N) X (cm) X (m)

.050 .49 5.7 .057

.100 .98 10.9 .109

.150 1.47 16.6 .166

.20 1.96 22.1 .221

.25 2.45 27.6 .276

.30 2.94 33.1 .331

.35 3.43 38.6 .386

.40 3.92 44.1 .441

Tapered Spring in Parallel

m(kg) F=mg (N) X (cm) X (m)

.50 .49 2.3 .023

.100 .98 5.0 .050

.150 1.47 7.7 .075

.20 1.96 10.5 .105

.250 2.45 13.3 .133

.30 2.94 15.5 .155

.350 3.43 18.3 .183

.40 3.92 20.9 .209

Tapered Springs in series

m(kg) F=mg (N) X (cm) X (m)

.50 .49 10.1 .101

.100 .98 19.7 .197

.150 1.47 30.4 .304

.20 1.96 40.9 .409

.250 2.45 52.7 .527

.30

.350

.40

Part 3: rubber Band

m(kg) F=mg (N) X (cm) X (m) X.5 (m)

.500 .49 1.3 .013 .114

1.00 .98 3.4 .034 .184

1.50 1.47 5.1 .051 .226

2.00 1.96 8.0 .080 .283

2.50 2.45 12.0 .120 .346

3.00 2.94 15.3 .153 .391

3.50 3.43 18.2 .182 .427

4.00 3.92 20.0 .200 .447

4.50 44.1 22.5 .225 .474

Calculations:

KCylindricalSpring: = 14.51 N/M

% Error=100% x [R2-1]/1= [.9998-1]/1 x 100%= .02%

KTaperedSpring1: 9.864 N/M

% Error=100% x [R2-1]/1= [.9999-1]/1 x 100%= .01%

KTaperedSpring2: =8.895 N/M

% Error=100% x [R2-1]/1= [1.00-1]= 0%

KParallelTheo= K1+K2 = 9.864 N/M + 8.895 N/M= 18.759 N/M

% Error=100% x [KParallelTheo-KParallelExp]/ KParallelTheo= [18.55-18.759]/ 18.55 x 100= 1.12%

KSeriesTheo: = 1/k1 + 1/k2 = 1/ktotal= 1/9.864 N/M + 1/8.895 N/M =4.68 N/M

% Error=100% x [KSeriestheo—KSeriesExp]/ KSeriesTheo = [4.68 N/M-4.673]/4.677 x 100 = .15%

% Error=100% x [R2-1]/1=[.9800-1]/1 = 2.00%

Rubber Band K=135.4N/m F=kxB B=0.7

Cylindrical Spring Integral: .078150 Joules

Tapered Spring Integral: .04352 Joules

Rubber Band Integral:1.303 Joules

KE = 1/2 mv2

(2(.07150)/.1).5=1.196m/s

(2(.04352)/.1).5 =.933 m/s

(2(1.303)/.1).5=5.105m/s

Discussion: In the first part of the experiment, the results indicate that the force and stretch have

a linear relationship. That is to say that if the force on a spring is applied in equal increments, the

distance in each successive stretch will be the same too. This is reflected in the obtained data as

well as in the graphs plotted from the obtained results. From the results in the cylindrical spring

and the tapered spring one and two, we can notice that the cylindrical spring is about twice as

tough to stretch. Also the springs that were stretch in parallel are stronger that when place in

series. When the spring was placed in series it lost about ¾ of its strength with respect to the

parallel results. The spring constants obtained from the experiment are precise with a 2 percent

range. The correlation values also indicate the precision of the results. The integral of the graphs

from x=0 to x=.1m is the energy stored in that particular spring from which the integral is

obtained. From the results is noticed that the cylindrical spring contains more potential energy

than the tapered spring. Also in the rubber band stretch we notice a square root relationship

rather than linear. In this experiment the rubber band is the one with the highest potential energy,

therefore if one wanted to use either the springs or the rubber band to project a paper clip, the

rubber band will have a better projection. In the results the rubber band has an initial potential

velocity of 5.105m/s while the cylindrical spring is 1.196 m/s.

Conclusion: The extension of the spring is proportional to the force in a direct proportionality.

All the springs tested in the experiment obeyed Hook’s Law, and the limit of elasticity was not

surpassed. In the rubber band experiment the result is not linear but it obeys a square root

relationship between the force and the stretch. A negative sing is placed on the spring constant K

because the restoring force is opposite to the displacement but in this experiment the force was in

the same direction as the displacement. The equation F=kx were F=mg gave the same

relationship when placing the springs in parallel and in series. The spring constant in the parallel

experiments behaves as capacitors in parallel. Mathematically is KTotal=K1 +K2 is analogous to

CTotal=C1+C2. By placing springs in parallel the force required to displace the spring will be

greater because the spring constant has increased as noted in the addition from the equation. In

the series spring experiment, the spring becomes weaker, also analogous to how resistors add in

series. This results in a lower resistance by placing resistors in series, in this case the spring’s

constant is what weakens needing a weaker force to cause displacement. When the spring was

stretched, the spring stored potential energy with the increment of force, the greater the force the

greater the potential energy stored, the energy stored by the spring is given by the integral of the

curve which is always nonnegative. One ought to be careful while doing this experiment since

one can get hooked on the astonishing behavior embedded in the physical world. As British

physicist would say Ut tensio, sic vis, which means, "As the extension, so the force".

Elasticity and Hook’s LawAlberto Arcea

Yambo Wang (partner)Physics 1 Lab (Thursday)

Mr. Kiledjian04/5/2011