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 ?
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?
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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