Swinging fun

D. Leela and classmates, Chennai

"Let's go to the beach," said Leela.
"Not again," said Maya.
"Any way it's too late now," said Nivedita. "We should have gone early in the
morning. It will be very hot on the beach now."
"Let's go tomorrow morning, then," said Leela, eagerly. 
"That sounds fine, but what shall we do now?" asked Nivedita.
Just then Mother came in with cool glasses of home-made lemon juice.
"Why are you all sitting around and not playing? It can't be that hot!"
"Aunty, we just finished a round of 'Sets' and we don't know what do
play next," said Nivedita. "Can we go out some-where?"
"Why not?" asked Mother. "Where do you want to go? Not too far," she
added hurriedly when she saw Maya's mouth opening on the word 'zoo'."
"I know," said Leela, "how about the Planetarium?" It's just the right
time to sit inside an air-conditioned room and watch the stars and night
sky in comfort!"
"You modern children," grumbled Mother, but not very seriously. "Always
wanting comfort. When I was young we used to go to the beach at night
and watch the night sky---the real one I mean, not in a theatre. But
there's something about the beach air at night: I used to always fall
asleep and never learned anything about stars!"
Every one laughed. "Come," said Leela, "here's your chance to find out
all about them."
It was evening. The children had finished their show and shivered
through the 3-D movie and the ice-cream. They were headed straight for
the Science park. 
"I'm going to the big swing," shouted Maya. "Me too," yelled both Leela
and Nivedita and the three ran off while Kavita walked slowly behind
with Mother. By the time the two reached the swings, all the three girls
were seated on a swing and shouting in joy. 
"I'm swinging faster than you," shouted Maya. "Bet you can't go as fast
as I can!" 
"Of course I can," said Leela. "I'm much stronger than you. Just watch."
She began to push violently on the ground, heaving the swing higher and
higher.
"You can't do it. Y-o-u c-a-n-'t d-o i-t!" shouted Maya in a sing-song
voice, as she kept nimbly swinging up and down while Leela was still
swinging out. Leela got red in the face as she huffed and puffed but
everyone could see that Maya was simply going much faster than the
others. 
"Leela, think," said Mother. "What about?" grumbled Leela. "This is a
Science Park right?" asked Mother quietly. "This is not just a
children's park with some swings in it!"
"Oooh," said Leela, on a long tired breath. "You mean it's a special
swing that will only go slow?"
Kavita smiled, "Nivu is going faster than you. And Maya is faster than
Nivu." Then she sat on one of the swings and began to push herself. "And
I'm going to be the fastest of them all!" And she was right. Maya, who
had been crowing over being the fastest, could only watch as Kavita
swung rapidly back and forth on her swing.
Leela and Nivedita both got off their swings and watched. Clearly Kavita
had understood something they hadn't.
Maya kept giggling. "You both look like someone put a spell on your
swings. Just accept it, you can't swing as fast as I can! Kavita's
bigger: only she can do it."
Kavita got off her swing and sat on the one Leela had just got off. She
pushed hard and began to swing. Maya's mouth dropped open when she saw
Kavita shoving the ground really hard but moving slowly, really slowly.
"Mother, someone's really done something to that swing," quavered Maya.
The others laughed, but not unkindly. "Think, Maya," began Mother. 
Just then Leela said, "I think I've got it." She pointed to the chains
holding up the swings and said, "Four swings, on chains of four different
lengths."
"Yes," said Nivedita. "And the longest swing takes the longest time to
go up and down!"
Maya got off her swing and looked up at the assembly of swings. "Kavita
was sitting on the shortest one. I had the next longer one. And Leela
was sitting on the tallest swing. Amma, look, that swing is so high,
higher than the tree!"
"Is that the reason, Aunty," asked Nivedita. Kavita nodded. She said,
"I've learned this in school, but I never thought it would be fun to see
on a real swing."
"Go on," said Mother as Kavita hesitated and looked up at her. "You seem
to know it all."
Kavita laughed self-consciously. "Well, actually in school we just
learned a formula. I don't know how to explain that to Maya!"
"We've come here so many times and never noticed this," exclaimed Leela.
Mother suggested, "Two of us are wearing watches. Why don't we find out
how much longer time the longer swing takes."
"That's a good idea," said Nivedita. "Aunty, I've been looking at these
swings and it appears to me that the third one is exactly twice as long
as the first."
"Very nice" approved Mother. "Shall we time these two swings?" 
Maya said, "Amma, I'll go and swing on each and you can count the time
it takes!" She ran and sat on the first swing and began to swing
furiously.
"Should I swing fast or slow?" she asked. Leela said, "first fast, then
slow."
So Maya puffed her way up and down on the swing 10 times.
"That took about 30 seconds," said Nivedita. "Now do it slowly." Maya stopped
the swing and gently swung up and down 10 times.
"That also took about 30 seconds," said Nivedita. "How can it be?"
Kavita said "We learned this too but I forgot it. The speed doesn't
depend on who's sitting. It has a natural swing frequency." 
"But I pushed so hard the first time!" said Maya. "Maybe it makes a
small difference but we can't see it on the watch," said Kavita.
Leela grew impatient. "Maya, now sit on the third swing."
Maya jumped off, saying, "Well, if it takes the same time I'm not going
to push hard! I'm going to sit back and enjoy the swinging and you can
time it!"
She swung 10 times up and down on this swing too. "Forty seconds," said
Nivedita. "That's not much slower. But if you can't catch up at all, I
guess it becomes pretty obvious that one is slower than the other."
"Yes," said Mother. "That's why Leela and Maya didn't notice it. If all
swings are moving then it's immediately clear that the longer ones are
slower."
The children went back to the swings. This time Leela made sure she took
the shortest one. Kavita was resigned to the longest one. After a time,
she said, "There's something very nice about this swing. It's slow but
it seems to move in such a flat manner that you are always close to the
ground. It's a very nice feeling."
"Yes, said Mother, "you are right. All the swings are moving forward and
backward but they are also moving up and down (because the chain-length
is fixed). You can see that Leela's swing is moving higher up than the
others. Yours is hardly moving up, because the chain is so long. You
won't easily find such a swing in a children's playground. Why, the
swing must be nearly 6 meters high! That's the height of a two-storey
building!"
"Yes," said Leela, stopping her swinging. "You are exactly right. On
the long swing there's no "going-around" feeling that I get on this
short swing. Here I feel like I'm on a ride in a merry-go-round; the
kind that goes up and around."
"That was nice, Aunty. Thanks!" said Kavita when they went back home.
Nivedita was lost in thought. She took a tenniquoit and began to swing
it around. 
"I can do a trick," said Leela. She held out her hand with palm upwards,
put a saucer on it and then swiftly turned the hand downwards and back
up again. The saucer did not fall. "Isn't this a nice swing? How nice it
would be if our swings don't stop but keep turning all the way around!"
"You can do that with a mug of water," said Kavita. "Take some water in
a mug, hold it well, turn it swiftly upside down and then turn it up
again at once. The water won't fall down."
"Why does that happen?" asked Leela. "Let's try it first," said
Nivedita.
They all rushed off to the bathroom to try it out. "Stay there," called
Mother from the kitchen. "Then if the water spills, it doesn't matter."
Soon laughter and giggles and squeals came from the bathroom. The
children were clearly not at work on a serious science experiment.
Why do you think it works? If you know the answer, or think you know the
answer, do write to us at JM, at the address given in this issue.

BOX
If you are curious to know Kavita's formula for the speed of a swing, it
can be calculated by assuming that a swing is just a simple pendulum.
That's the little object that swings up and down on old-fashioned
clocks. If you measure the length of the swing in meters, then the time
taken to go from the middle (starting position) to one end (either
forwards or backwards) is given by the square-root of the length. So a 4
meter swing will take sqrt(4)=2 seconds to swing from the middle to the
front. Usually, the time-period is considered as the time taken to start
from the middle, go fowards, come back, go backwards, and come back to
the middle again (one full cycle). So the time period of a 2 meter swing
is 4 seconds.
The exact formula is time period T in seconds is given by
T= 2pi sqrt(l/g) where g=9.8 m/s/s is the acceleration due to earth's
gravity. If you work this out, you will get T/2 = sqrt(l) as a very good
approximation.
Note: T^2 = l; can you see the parabola here?!