GRAVITY AND FALLING BODIES
Gravity is one of the most familiar forces in nature; its effect on motion has been a subject of discussion for centuries. If an object is dropped from a great height, it can be observed that it falls with ever increasing speed until air resistance balances the effect of gravity, at which time it is said to have reached its terminal velocity. The term free falling bodies is used for objects that are moving freely under the influence of gravity, whether they are moving upward or downward. Any object that has no forces other than gravity acting on it is said to be in free fall, whether it is moving upward, downward, or in any direction.
It is found that if air resistance can be made negligible, then falling bodies will accelerate toward the center of the earth at the same rate, regardless of their mass. The value for the acceleration of gravity, given the symbol g, has been measured on earth as g = 9.8 m/s2. Galileo was the first to demonstrate that all bodies fall at the same rate if air resistance is negligible. ( It is often said that he did this by dropping objects of various masses from the Leaning Tower of Pisa, although there is no historical evidence that he actually used the famed tower.) Galileo’s recorded experiments settled some very old controversies about falling bodies, proving less-popular ideas to be correct.
Even more important than his discoveries about falling objects was his breaking away from old methods of determining truth. Galileo is often credited with being the Father of Modern Science because of his forceful demonstration of the value of observation and the discoveries he made through his ingenious experiments.
The following is a data from one of Galileo’s earliest experiments of a ball rolling down an inclined plane. His data were recorded on his notes. Galileo held a ball at the top of an inclined, grooved board and marked its position. Releasing the ball, he marked its position at the end of equal intervals of time. This is much like dropping a ball from a height, except that the effect of gravity has been “reduced” by allowing the ball to roll slowly down the inclined board rather than falling straight down. The position as measured by Galileo are given in the following table :
Time t (equal intervals)
|
t2
|
Distance S (points)
|
S/ t2
|
1
|
1
|
33
|
33.0
|
2
|
4
|
130
|
32.5
|
3
|
9
|
298
|
33.1
|
4
|
16
|
526
|
32.9
|
5
|
25
|
824
|
33.0
|
6
|
36
|
1192
|
33.1
|
7
|
49
|
1620
|
33.1
|
8
|
64
|
2104
|
32.9
|
The observations show what was already known quantitatively to Galileo and others of his time – that a rolling (or falling) object picks up speed as it continues to roll (or fall). However, the debt we owe to Galileo is for his careful measurements and his quantitative (mathematical) interpretation of the data. His object was to find a general rule describing how distances increase with increasing time of fall. After some trial and error, and with considerable insight, Galileo realized that the distance traveled was proportional to the square of the elapsed time.
S at2 ==> S = h = ½ at2
Problems
1. A ball is thrown vertically up
with an initial velocity of 15 m/s. How high does the ball rise from its
projection point ? How long does it take for this ball to reach the highest
point. How high does it go in 2 seconds ? in 3 seconds? What is the time
required to travel a height of 9 m?
5 m?
5 m?
Ans. ( 11.48 m
, 1.53 s , 10.4 m , 0.9 m , 0.82 s , 2.24 s , 0.38 s
, 2.68 s )
2. A rock is dropped from a bridge
55 m high relative to the water of a river below. How long will it take for the
rock to reach the surface of the water ? Calculate the positions of
the rock 0.5s, 1.25 s, 2s after it was release relative
to the water.
3. A metal sphere is dropped from a
50 m high tower. Determine the height traveled by the sphere in the time
interval from 0.25 s to 1.25 s.
4. "Khalifa Tower", pronounced in English (/ˈbɜrdʒ kəˈliːfə/), known as
Burj Dubai before its inauguration, is a skyscraper in Dubai, United Arab
Emirates. It is the tallest man-made structure in the world, standing at 829.8
m (2,722 ft). If an object
falls from the top of this tower, how many seconds does it reach the
ground ? What is the height travelled by the object in the interval
between 6 seconds and 10 seconds after falling?
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