Pendulum Lab report Physics 141 Lab

Pendulum Lab report

Physics 141 Lab

Abstract

In this experiment a pendulum’s period was measured and those values were used to find gravitational force acting on the pendulum g. This value g was then added and suptracted from the standard deviation of the trials taken during the experiment and compared with the literature value of 9.793 m/s^2. The g values found in the experiment were: Therefore g=9.1756 m/s^2 + 0.0122(standard deviation)   =9.1878 m/s^2 and g=9.1756 m/s^2 – 0.0122(sd) = 9.1634 m/s^2. When compared with the literature gravitational constant value the percent accuracy was found to be 93.8% and 93.6%.

Goal of the experiment

The goal of this experiment is to find the gravitational force in Tucson acting on a pendulum as it swings. This will be found using variables that are measurable on the pendulum such as period and length (height of pendulum). This is a useful experiment because it allows for practice and knowledge of the theory behind pendulums as well as determining how the gravitational constant that is used in solving common physics problems of 9.8 m/s^2 is not applicable at all elevations and allows for knowledge of the gravitational constant acting on you where you live.

Introduction

For the execution and completion of this lab an equation is needed that relates gravitational force constant g to period (T)[time it takes to swing back and forth one time] and Length/Height (L) [in this case two ends of a string are tied to a rod making a parabola shape with the string and the height is measured from the minimum extrema of the string to the rod with which the ends of the string are tied to]. This formula will be derived in the theory.

Theory:

Periodic Pendulum

||T||=constant (Tx^2+Ty^2)^(1/2)

Fnet=mgsin(theta)

M*acceleration=-mgsin(theta)

Using approximation that sin(theta)=theta for angles less than 10 degrees.

Such that mass*acceleration  -mg*theta

x=L*theta

A=second derivative of x/second derivative of t

X=L*theta

L=second derivative of theta/second derivative of time

M*L*d²theta/dt²=-mg*theta

D²theta/dt²=theta₁*sin(w*t+Φ)

-m*L*theta₁*w²*sin(wt+Φ)=-mgtheta₁*sin(wt+Φ).

L*w^2=g

W=sqrt(g/L)=2pi/T 

G=(2pi)^2*L / (T^2)

Procedure

• Measure length from rod which the string is tied to, to the bottom of the string where the weight is hung.
• Set up Explorer to find the time between gates
• Starting from rest each trial. Swing the pendulum at about the same period (don’t vary between tiny pushes of the pendulum and really large pushes of the pendulum). And measure the two times with which the Xplorer seems to alternate between.
• Find the average of these times, and find the average of the average of the times with previous averages until the times converge about a point.

Results

L=34.3 cm or 0.343 m

Trial	Times	Average time	Average	g	SD (g)
1	0.625628	0.58683	0.606229	0.606229	9.210747	0.012200567
2	0.623572	0.59316	0.608366	0.607298	9.178364	Average (g)
3	0.626892	0.588498	0.607695	0.60743	9.17436	9.175614599
4	0.62095	0.595818	0.608384	0.607669	9.16716
5	0.621776	0.59339	0.607583	0.607651	9.167676
6	0.625552	0.583988	0.60477	0.607171	9.182184
7	0.622508	0.591632	0.60707	0.607157	9.182621
8	0.621322	0.597094	0.609208	0.607413	9.17487
9	0.624928	0.58877	0.606849	0.60735	9.176764
10	0.62709	0.585404	0.606247	0.60724	9.180099
11	0.625738	0.589022	0.60738	0.607253	9.179714
12	0.624946	0.58839	0.606668	0.607204	9.181188
13	0.62131	0.600594	0.610952	0.607492	9.172476
14	0.621802	0.60349	0.612646	0.607861	9.16137
15	0.622492	0.594492	0.608492	0.607903	9.160101
16	0.626738	0.589022	0.60788	0.607901	9.160143

Calculations

Using the final time that the period converged to the gravitational constant can be found using the results.

According to the theory gravitational force g is found by g=(2pi)^2*L/(T^2) however one  slight modification needs to be made. Because the time between the gates was measured only the time was measured for it to complete half a period so the T needs to be multiplied by 2. Giving:

G=(2pi)^2* L (m)/(2*T(s))^2

G=((2*3.1415)*0.343 m)/(2*0.607901 s)^2 = 9.1601 m/s^2 (Example)

According to the lab notebook g=<g>+,- standard deviation (g)

Therefore g=9.1756 m/s^2 + 0.0122   =9.1878 m/s^2

                g=9.1756 m/s^2 – 0.0122   = 9.1634 m/s^2

The gravitational constant given in lab to use for theory at this elevation is 9.793 m/s^2

Percent accuracy= experimental/theory * 100

% = 9.1878 m/s^2 / 9.793 m/s^2 *100 = 93.82 % accuracy

% = 9.1634 m/s^2  9.7934 m/s^2 * 100 = 93.57 %accuracy

Which leaves about 6-7% error.

Discussion

*What was your final result for g?

Final result for g was

Average value of g was g=9.1756 m/s^2

g+1 SD = 9.1878 m/s^2

g-1 SD = 9.1634 m/s^2

Percent accuracy was 93.82% and 93.57% using the SD corrected values.

*What equations did you use to analyze the data and where do they come from?

The formula derived in the theory used to calculate g was g= ((2pi)^2*L)/(2T)^2

This formula was derived from Newtons second law sum of net forces = mass times acceleration using the forces acting on the pendulum system when at various positions along its oscillation. Some of the assumptions made were that sin(theta)=theta for values less than 10 degrees. Additionally the formula for motion of a harmonic oscillator which is X(t)=A(t)*cos(wt+phi) and subsequent first and second derivatives of that formula was used where A is the amplitude, phi is the starting position at equilibrium and w is the angular frequency of 2pi/T.

*What are the sources of error in your experiment and how did they affect the data?

Some sources of error in this experiment would be

• Not accounting for friction between string tied to bar (system error). This would slow the pendulum down by a fractional amount in comparison with the ideal system used in theory.
• Not being able to swing the pendulum with the exact same force every time (random error). Not able to repeat every case identically. The average was taken to reduce this error but there is still a variance between found g value and the literature value.
• Using the assumption that sin(theta)=theta (even though small angles were used there would still be some error from that. (system error). E.g. the gravitational value found should be smaller than the literature value due to this approximation.

*Did you use a long or short pendulum? Large or small angles? Why?

A medium sized length pendulum was used so that it would be easy to manipulate so the forces of the push would be easily repeatable so the data would be as accurate as possible. *although the string still had to be long enough such that the mass would be able to cross the gate sensor.

Small angles were used because of the approximation in the theory that allowed for the derivation of the equation so the pendulum was not pushed really hard making the pendulum to swing out really wide but rather light pushes were given to keep theta small, and reduce error.

*If you were to do this experiment again, what would you do differently and why?

If I could do this experiment again I would probably run the experiment twice with varying weights. The mass value is not used in the formula so I would like to verify that by using different masses on the pendulum and seeing how the results from that looked like.

If I were to do this experiment over again with the goal of reducing error I would set up the experiment using one string with one end tied to the weight and the other to the bar instead of both to the bar with the weight in the middle to reduce the friction that the string and the bar had with each other and to make it easier to determine L because there would be no possibility for the pendulum to move up the sides of the strings by a fractional amount.