After the excitement of Roger’s cup tennis matches and particularly watching the match between Djokovic and Monfils at the center court, it is time to get back to Galileo’s experiment.
In the last post, we looked at the life of Galileo, his contributions and the punishment by church for his discoveries.
At the time of Galileo, the laws that govern the motion of bodies were thought to follow the ‘Aristotelian’ beliefs.
Aristotelian Laws of Motion
Aristotle was a great greek philosopher and also a teacher to Alexander ‘the Great. He lived in 340 BC and had established a university called ‘Lyceum’ where objects and natural ph
enomenon were studied and theories formed
He had great belief in geometry and his university had a sign ‘Let No One who cannot think geometrically enter here’
His laws of motion were based on observation of everyday objects and activities like falling of a stone/feather through air/ water, pulling of a cart by a bullock, movement of stars and planets etc. Some of these laws are listed below:
- He believed that all things are made from combination of earth, water, air , fire and their characteristics determine the motion of various bodies.
- Earth and water move downwards but earth move with greater force which can be observed by a stone sinking in water.
- Air and fire move upwards with fire rising faster which can be observed by hot air rising above cold air.
- Similarly heavier things fall faster and their speed increases with the weight of the object. Therefore a stone falls faster than a feather.
- Speed of fall varies inversely with the density of the medium, an objet will fall twice as fast through a medium half the density. Therefore an object will fall faster in air than in water.
- Speed of a moving object is proportional to the applied force like that a bullock pulling a cart. If you stop pushing, the body stops moving.
- Circular motion of planets could not be explained by these laws and therefore he believed that planets are not made of the four earthly elements but a fifth element ‘ether’ whose natural motion was circular
Aristotle’s laws were merely based on observation and were not verified by experiments.
These ‘Aristotelian laws’ remained unchallenged for about 2000 years till Galileo proved them incorrect by ingenious experiments. However the concept of ether continued till 1887 when Michelson-Morley through their experiment proved the absence of ether.
During this post, we will discuss Galileo’s experiments and Michelson-Morley experiment separately in a subsequent post.
Galileo’s Laws of naturally accelerated motion
As seen in the description of Galileo’s life, his curiosity to examine the motion of objects was aroused when he saw a swinging pendulum. Though this experiment is not part of the ‘Best Experiments’, it is necessary to examine it because this started a search leading to his more famous experiments and laws.
Experiments with Pendulum:
You all must be familiar with ‘Pendulum’ experiment. In this experiment, a weight is hung from a tripod using a thread. The weight is pulled to one side and then released. It starts swinging left to Right to Left. Using a timing clock, time for number of swings are measured.
The experiment is repeated for different weights, lengths of thread and the angle of swing. Based on the results, it is observed that the change in the weight does not affect the time of swing but is affected by the length of the thread.
This is the law which Galileo had first discovered and was to seriously impact the ‘Aristotelian’ beliefs held for last 2000 years.
In 1581, at the age of 20, he saw a chandelier swinging in a church at Pisa. He counted the number of swings using his pulse as a timing
device. He was surprised to find that the time of swing is independent of how far it moves (wider or narrower).
He carried out more experiments and established laws of Pendulum. The laws were:
- It takes same time to swing from side to side whether the swings are wider or narrower
- It takes same time to swing from side to side however heavy it is
- Time of swing changes with length of pendulum.
Designing a clock:
Galileo realized that pendulum due to its constant time of swing can be used as a timing device. He suggested that the patients pulse rate could be measured by such a device. But Galileo did not realize that the timing is accurate only for small angles of swing. This was corrected by a Dutch scientist, Christiaan Huygens in 1656 who built an accurate clock using pendulum
As an important part of better understanding of the principle of pendulum, let us look at the formula that governs the movement of pendulum.
Key questions, equations & parameters:
First question that arises is
‘why does the pendulum swing ?’
I am sure you are aware with the potential energy (PE) and kinetic energy (KE).
When we pull the weight of the pendulum to say left side, the weight is raised above its center position. Thus it acquires ‘Potential Energy’.
When the weight is released, it starts moving to the lower position, thus converting its PE into KE. This KE carries it to other side and stops at a certain height depending on its KE.
This keeps on repeating till all its energy is lost and it returns to original center position. It is important to realize that the energy is lost due to friction of the air and at the fulcrum of the thread. If this resistnce was zero, it will continue to swing for ever !!!
Formula
The formula that governs the law of pendulum is expressed as:
- time of swing t = 2π√(l/g)
where t is the time of swing in secs, l is length of the thread in meters, and g is the gravitational acceleration = 9.8 m/sec^2 (usually taken as 10m/sec^2 to simplify calculations)
This equation tells us that the time of swing will change directly as under-root of length of string and inversely as per g.
Thus to get higher time of swing, and thus better accuracy, length of suspension need to be increased.
Moreover the time of swing will change if we are on moon.
- On earth where g = 9.8 m/sec^2 (usually taken as 10m/sec^2 to simplify calculations)
time of swing t = 2 x 3.14 x √l x 1/3.13 ≈ 2 √l
We can extend this further by examining the factors that influence the length of the string.
Factors affecting the length of string:
Length of a string is influenced by the material of which it is made and also its expansion with change in temperature. This is represented by the formula:
- Length of swing at temp T = L = Lo (1+α*T)
where T = temp rise in deg c; α = temp co-eff of the string
- Length of swing at temperature T1 = L1 = Lo (1+αT1);
- and length at temperature T2 = L2 = Lo (1+α*T2)
Therefore Time of swing t at temp T2 = √(L1/L2)
This can be used to calculate time lost in one day
(t – to)/to = ½ α T * 86400 secs
Exercises:
Ques 1 – To keep correct time, modern watches are fitted with a balance wheel made of which material:
a) steel b) platinum c) Invar d) tungsten
Ans: c) Invar – it is an alloy of steel-lead and has lowest temp co-eff (1.2ppm/C)
Ques 2 – A clock keeps correct time at 25 deg C and has a pendulum made of a material having a temperature co efficient of 19 ppm/C. How many secs will it gain if temp is 0 C !!!
L25 = 1.000475 Lo; T25/T0 = √L25/Lo = 1.000237; (T25-T0)/T0 = 0.000237;
gain in time for one vibr = 2x.000237; 1vibr = 2 sec; no of vibr per day = 24x60x60/2 ;
Time gained = 2x.000237x24x60x60/2=20.52 sec
I hope this gives you a better understanding of a pendulum much above in your class room/ text book.
Personal verification
Do carry out your own experiment using a pendulum and check for yourself the variation in readings that you may get.
Read earlier post of July 21 for guidelines of carrying out experiment.
While carrying out the experiment, explore various variables and their effect …. do not go with pre-conceived ideas … your results should speak for themselves because even Galileo was not fully correct in his observations which were corrected by Huygens.
Links
There are also many good websites explaining the pendulum experiment.
Here is a note from http://www.worsleyschool.net which you may find useful. This also explains the concept of variables in experiments. I am unable to locate this site now but here is a link to the document copied from that site earlier:
https://drive.google.com/file/d/0B2NS0FqvGiDJX1BVeGlGWmR6N0U/edit?usp=sharing
S4S-Science4Students 