Of A Simple Pendulum Essay, Research Paper

Investigating factors which affect the period time of a simple

pendulum Planning Definitions: Oscillation : Repeated motion of pendulum (to and for)

Period (T) : Time taken for one full oscillation In this investigation, I am going to experimentally determine a factor which

will affect the period of a simple pendulum and the mathematical relationship

of this factor. This type of pendulum will consist of a mass hanging on a length

of string. Factors which affect the period (T) of a pendulum:

-Length (L) of pendulum

-Angle of amplitude

-Gravitational field strength (g)

-Mass of bob I predict that the period will be affected by the length of the pendulum. An

increase in length will produce an increase in time. I based by prediction on

the scientific theory I found in a physics text book: The pendulum is able to work when the bob is raised to an angle larger than

the point at which it is vertically suspended at rest. By raising the bob, the

pendulum gains Gravitation Potential Energy or GPE, as in being raised, it is

held above this point of natural suspension and so therefore is acting against

the natural gravitational force. Once the bob is released, this gravitational

force is able to act on it, thus moving it downwards towards its original

hanging point. We can say therefore, that as it is released, the GPE is

converted into Kinetic Energy (KE) needed for the pendulum to swing. Once

the bob returns to its original point of suspension, the GPE has been totally

converted into KE, causing the bob to continue moving past its pivot point and

up to a height equidistant from its pivot as its starting point. The same factors affect the pendulum on its reverse swing. GPE gained after

reaching its highest point in its swing, is converted into KE needed for it to

return back to its natural point of vertical suspension. Due to this continuous

motion, the bob creates an arc shaped swing. The movement of the pendulum

is repeated until an external force acts on it, causing it to cease in movement.

The pendulum never looses any energy, it is simply converted from one form

to another and back again.

I am therefore going to experimentally determine the relationship between the

length of the pendulum and the period.

In the scientific theory, I found a formula relating the length of the pendulum

to the period. It stated that: P = 2 L

g P = The period

g = Gravitational Field Strength

L = Length of string This formula shows that L is the only variable that when altered will affect the

value of P, as all the other values are constants. The formula:P = 2 L

g can be rearranged to produce the formula: P = 4L

g and therefore:P = 4

L g As 4 and g are both constants, this means that P must be directly

proportional to L. I can now say that the length of the pendulum does have an affect on the

period, and as the length of the pendulum increases, the length of the period

will also increase.

I will draw a graph of P against L. As they are directly proportional to each

other, the predicted graph should show a straight line through the origin:Method

-I will firstly set up a clamp stand with a piece of string 50cm long

attached to it.

-A mass of 50g will be attached securely to the end of the string

-The mass will be held to one side at an angle of 45 degrees (measured

with a protractor), and then released.

-A stop clock will be used to time taken for one full oscillation

-This will be repeated a number of times, each time shortening the

length of string by 10cm

-The length of the pendulum will be plotted against the period on a

graph. NB. The final length of string and mass will be decided after my preliminary

investigation. Apparatus:

-Meter ruler

-Protractor

-Clamp stand

-G-clamp

-Stop clock

-String

-Mass Diagram: The following factors will be considered when providing a fair test:

-The mass will be a constant of 50g throughout the experiment

-Angle of amplitude shall be a constant of 45 degrees. This will ensure

that there is no variation of the forces acting on the pendulum.

-The value of gravitational field strength will inevitably remain constant,

helping me to provide a fair test.

-The intervals between the string lengths will increase by 10cm each

time. This will help me to identify a clear pattern in my results.

-If any anomalous results are identified, readings will be repeated. This

will ensure that all readings are sufficiently accurate.

-To ensure that the velocity is not affected, I will ensure that there are no

obstructions to the swing of the pendulum. The following factors will be considered when providing a safe test:

-Care will be taken not to let the bob come into contact with anything

whilst swinging the pendulum, as the weight is relatively heavy (50g)

-The clamp stand will be firmly secured to the bench with a G-clamp so

that the clamp stand will not move, affecting the results.

-Excessively large swings will be avoided (angle of amplitude will be 45

degrees Results of preliminary investigation: Length of string (cm)Period (secs) 502.58 402.31 302.11 201.78 101.39

My preliminary investigation was successful. The results from my table back

up my prediction that, as the length of the pendulum increases, the period

increases. I learned from my preliminary investigation that my proposed method may

not give me sufficiently accurate results. These results may be inaccurate due

to a slight error of measurement in time, height or length. Although this

experiment produced no anomalies, I will take three readings of each value

during my final experiment and take an average. I will also measure the time

taken for 5 oscillations rather that 1 and then divide the result by 5. These

two changes will hopefully help me to identify and eliminate anomalies,

should they occur. They should also add to the accuracy of my results. Obtaining Evidence I used the method proposed in my plan, taking three readings of each value

and measuring the time taken for 5 oscillations rather than for 1. During the

experiment, I observed that each oscillation for the same length of string

seemed to be equal. This showed that the pendulum did not slow down as the

number of oscillations increased. I took the safety measures described in my

original plan. During the experiment I was careful to use accurate measurements in order to

obtain sufficiently accurate results, for example:

-The string was measured with a meter ruler, to the nearest mm, to

ensure that each measurement had a difference of exactly 10cm.

-The angle of amplitude will be measured with a protractor to the

nearest degree to ensure that the angle remains constant throughout

the experiment.

-A stop clock will be used to measure the period accurately. The period

was measured in seconds, with the stop clock measuring to the degree

of two decimal places of a second. However, I have rounded up each

time to the nearest second to give appropriate results.

-The mass was measured using five10g masses, to ensure that the mass

remained constant throughout the experiment. Results:

Length of string (cm)Period (secs)

507.2

8.1

6.45 406.25

6.6

6.4 305.6

5.2

6.15

204.55

4.5

4.6 102.95

3.25

3.0 I took three readings of each value and took an average for each concentration.

I then divided by 5 to get the average reading for one oscillation. This again

should influence the accuracy of my results. Table of averages:

Length of string (cm)Period (secs)

501.45 401.28 301.13 200.91 100.61 Using the formula, T = 2 L

g

found in the Scientific Theory, I calculated the perfect results that should have

been obtained, had my experiment followed the formula exactly:

Length of string (cm)Period (secs)_

501.44 401.25 301.07 200.91 100.64

Using my averaged results, I squared P to show the relationship between P

and L: Length of string (cm)Period (secs)_

502.1 401.64 301.28 200.83 100.37 As all my results were accurate, I had no need to repeat any of them. However,

had there been an anomalous result, or had I come across any problems, I

would have repeated my results to identify the cause and eliminate anomalies.

Analysing evidence and concluding Using the results from my table, I drew a graph to show what had been

obtained from the experiment (see graph A). The graph clearly shows a

smooth curve with a positive gradient. This indicates that as the length of the

pendulum is increased, the period will increase. Although my second graph (see graph B), does not show a perfect straight line

through the origin, a line of best fit can be drawn to show this. This backs up

the theory in my scientific knowledge, that P is directly proportional to L,

i.e. if the length of string was doubled, the period would be doubled. My table of results drawn from my experiment was extremely similar to the

results produced from the scientific formula, showing that my experiment was

successful. My two graphs showed resemblance to my predicted graphs,

indicating that my results were sufficiently accurate and therefore, my

proposed method was reliable for this experiment. My findings indicate that the time period varies directly with the length of the

string when all other factors remain constant.

Evaluating The evidence obtained from my experiment supported my prediction that as

the length of the pendulum increases, the period increases. This is also shown

in Graph A, as the graph displays a smooth curve with a positive gradient. My

method in squaring P was successful, as I discovered that T was directly

proportional to L, providing all other values remain constant. This was shown

by a straight line going through the origin (Graph B). These results were

encouraging and led me to believe that my proposed method was sufficient for

the experiment. Some of the results were not accurate, as they did not match the results

produced by the formula. This could have been due to human error. However,

the majority of my results were no more than a decimal place away from the

formula results and, therefore, quite reliable. Had there been any anomalous

results, I would have repeated my readings. Factors which may have affected the accuracy of my results include:

-Error in measurement of angle of altitude. This angle proved difficult to

measure and it was hard to get the exact same angle for each result. To

improve the accuracy of this measurement, I could have attached the

protractor to the clamp stand so that it was in a fixed position.

-Error in measurement of string. To improve the accuracy of this, I

could have marked off the points with a pen to ensure they were as

accurately measured as possible.

-Human reaction time. Depending on human reaction time, the

measurement period time could have been measured inaccurately, due

to slow reactions when setting the stop-clock etc. This could have been

improved by involving another person to aid me with my experiment,

for a quicker reaction time. The procedure was relatively reliable, excluding human error, and so I can

conclude that my evidence is sufficient to support a firm conclusion that: The only factor which affects the period of a simple pendulum is its

length. As the length increases, so does the period. If I were to extend my investigation, I would investigate to provide additional

evidence to back up my conclusion, for example, changing the mass or angle

of altitude. The results gained would hopefully aid me further in supporting

my Scientific Theory. It would also be interesting to investigate how the

factors are affected when the Gravitational Field Strength is different, i.e.. not

9.8 Newtons.

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