The maximum displacement of the body from its rest position is called amplitude. Angular amplitude is the angle between the extreme (highest position) and the rest position of the string and provided the angle is small, it can only be affected by the length of the pendulum, not the mass on the end, and the acceleration due to gravity. The energy used in the oscillation is gradually converted into heat energy. The oscillation’s amplitude decreases until the pendulum ceases motion at the point of equilibrium. When the amplitude is increased, the GPE is enlarged because the height is increased. Energy is converted from one form to another, it is never destroyed by the pendulum, meaning kinetic energy increases by the same amount as GPE.
When a pendulum swings to and fro, its energy is constantly changing from potential energy to kinetic energy and back again. This energy is gradually transferred to heat by friction with the air. This heat is ‘low-grade’ energy and we cannot make use of it.
This diagram shows the constantly changing energy of a pendulum:
This formula demonstrates that length affect the period of the pendulum:
t = 2π √ 1/g
where: t = the period of one oscillation (seconds)
l = the length of the pendulum (centimetres)
g = the strength of the gravitational field (10m/s on earth)
l is the only variable that is capable of affecting the result of the period as the other variables are constant.
When the pendulum is stationary:
T = mg = W
where: T = tension
m = mass
W = weight
G = the force of gravity on the Earth’s surface
When the pendulum is oscillating:
T = W, consequently T – W = 0 = Fres
Where: T = tension
W = weight
Hypothesis.
Due to the fact that length has an affect on the pendulum, I predict that as the length of the pendulum increases, the length of the period of the pendulum will also increase. The mass of the suspended body will have no affect on the time it takes for the pendulum to swing as long as the angle of amplitude ( the angle between the pendulum’s rest position and highest movement) is kept at a small angle.
Using the equation t = 2π √ 1/g I will calculate how long it will take for each length to complete 30 oscillations (the amount of oscillations I am using in the results) and these are shown in my table of results, in the predicted period column.
Plan.
My plan is for the investigation to test my hypothesis. The only variable I am investigating is the length of the pendulum’s string, and I will allow this to vary to the lengths I am investigating, which are (in metres):
- 0.8m,
- 0.9m,
- 1m,
- 1.1 m,
- 1.2 m and
- 1.3 m
The length is defined as being from the knife-edge to the centre of gravity of the suspended body.
The following variables will remain constant throughout the investigation:
- The mass of the suspended body,
- The period for one complete oscillation and
- The angle of release (constantly 5°)
The length of the string will be measured at the various lengths, starting from the largest, prior to being connected to the pendulum, and will be changed systematically. The pendulum will stopped being timed on a stopwatch after 30 oscillations and each length will be timed on 6 separate occasions.
The same mass will be used throughout the experiment, there will be no circular swings and no obstructions in the pendulum’s path. Draughts and tremors will be reduced to a minimum.
Preliminary Tests:
For a Key Stage 3 experiment, I investigated the affect of the mass of a suspended body on the period of the pendulum. Here is the table of results, which confirms that mass has no influence on the period.
To eliminate human and experimental errors, as well as errors gained from apparatus and to get accurate results, these measures must be undertaken:
- Using the hard bone, as opposed to the fleshy part, of the thumb to start the stopwatch to achieve accurate timekeeping,
- Accurately measuring the length of the pendulum’s string and
- Avoiding parallax (which occurs when the angle of the object is changed as a result of the position of the experimenter)
Apparatus:
- Recording equipment,
- Metre rule accurate to 1mm,
- A small, aerodynamically shaped and heavy body (2-4 oz),
-
Electronic stopwatch measuring to 100th of a second,
- Thin, inextendable string,
- G-clamp (for stability in draughts),
- Clamp stand (for stability in draughts) and
- A protractor
Set up as shown:
Results:
(See also graphs).
Conclusion:
The results I have obtained from my results show a definite increase in the time it takes the pendulum to do a complete swing when there is an increase in length. Therefore I can definitely say that I have found that an increase in length of a pendulum caused an increase in the time it takes for the pendulum to do a complete swing. This is due to the fact that only length and the acceleration due to gravity can affect the period of the pendulum. The results obtained agree with my hypothesis (that as length of the pendulum increases, period increases) and is shown by the clear set of values on the graph that increase as the length increases, producing a positive correlation. The positive gradient indicates that as the length of the pendulum increases, the period also increases. I have found that length is the only factor that affects the period of a simple pendulum is the length of the pendulum. This means that as the length of the pendulum increases, so does the oscillation period.
Evaluation:
The quality of my experiment must be good, as the results and conclusion do not contradict each other and they both agree with my hypothesis ‘‘as the length of the pendulum increases, so does the oscillation period’’. The majority of my data is consistent and the results obtained are close to the predictions I made for the period of the pendulum at different lengths, for example for the pendulum length of 1 metre, the predicted period was 2.09 seconds and we recorded an extremely close 2 seconds in the actual experiment, which meant that there were only slight experimental errors.
I trust that my method was successful as it provided me with accurate results, which correspond with the formula stated.
I found it hard to measure the angle of amplitude and this could be one reason why the two sets of results did not perfectly match. Other reasons for errors could include inaccurate measurements of the string and angle, and error in judgment when stopping the stopwatch. To eliminate these mistakes, a protractor could be attached to the clamp to measure the angle and more people, or perhaps a laser beam, could be used to gain a more accurate average of the time of the oscillations.
The only factors that affect the pendulum are the gravitational field of the Earth and the length of the pendulum. To extend the investigation, I could have timed each length more than 6 times and the average would have further ironed out any human errors. It may me useful to do the experiment again in a different place on Earth to see what affect the gravitational pull has on the pendulum. To investigate the work that has been started further, I could extend the prediction that I have made which would then need to include a bigger range of length variables.
