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Understand the exponential dependence of the height of a water column as it flows uniformly under the influence of gravity.

Extracts from this document...

Introduction

Tsering Norbu

Partners: Rahul, Takayuki, Pavan

A – Block

22nd January 2004

Expt.3Exploring a log-linear relationship.

Aim: To understand the exponential dependence of the height of a water column as it flows uniformly under the influence of gravity.

Theory: The height of water, h, of a water column in jar decreases exponentially in time, t, if the flow is continuous; i.e. the relationship is given through

        h(t) = h0 ∙ e k t,                                                 (Eq. 1)

where h0 is the height of the water at time t = 0, and k is some constant. By plotting h as a function of t, one obtains an exponential graph. By taking the natural logarithm of both sides of Eq. 1, we obtain

        ln(h) = ln(h0) – k t.                                                (Eq. 2)

Hence, if one plots ln(h) versus t, one obtains a straight line graph with ln(h0) as the y-intercept and k as the slope.

...read more.

Middle

Height (h)

 ± 0.1 cm

Difference in height           ± ∆ z cm

1

41.6

2

38.4

3.2 ± 0.1

3

35.2

3.2 ± 0.1

4

32.3

2.9 ± 0.1

5

29.4

2.9 ± 0.1

6

26.4

3.0 ± 0.1

7

23.5

2.9 ± 0.1

8

21.1

2.4 ± 0.1

9

18.6

2.5 ± 0.1

10

16.6

2.0 ± 0.1

11

14.5

2.1 ± 0.1

12

12.6

1.9 ± 0.1

13

10.9

1.7 ± 0.1

14

9.6

1.3 ± 0.1

15

8.0

1.6 ± 0.1

16

6.9

1.1 ± 0.1

17

5.7

1.2± 0.1

18

4.6

1.1 ± 0.1

19

3.8

0.8 ± 0.1

20

3.0

0.8 ± 0.1

21

2.4

0.6 ± 0.1

22

1.8

0.6 ± 0.1

23

1.4

0.4 ± 0.1

24

1.0

0.4 ± 0.1

25

0.6

0.4 ± 0.1

26

0.3

0.3 ± 0.1

27

0.2

0.1 ± 0.1

28

0.0

0.2 ± 0.1

The Graph of height of water column against time.

image00.png

(Table 1.2)                                                        (Graph 1.1)                

Interpretation

The table 1.2 gives the difference in the heights of successive readings. We can see that there is a decrease in the difference in the height of the successive readings.

It is also illustrated by the graph 1.1 which is an inverse exponential curve. Thus, rate of change

...read more.

Conclusion

0

ln(½h0) = ln(h0) – kt

ln(h0) – ln(2) = ln(h0) – kt

ln(2) = kt

t = ln(2)/ k

and let t = T½

CONCLUSION
I didn’t obtain a log linear relationship as stated in the lab sheet since we can see from the results I got from the graphs.

But there is a log linear relationship existing.

Evaluation

#

Theoretical prediction

Experimental results

Reasons

½h0

20.8 ± 0.1 cm

21.1 ± 0.1 cm

The big time interval between each of the readings prevented us from having better acccuracy

-38.5 ± 3.6 sec

70 secs

The time interval was too big.

In Graph 1.2 the value for the time at which the height has reached half of the initial height i.e. ½h0 which is theoretically equal to 20.8 cm but in my graph the value I can get closest to is 21.1 cm. This is quite accurate to the theoretical value. And the time is 70 seconds which is 70/270 roughly ¼ of the time taken to drain out all the water in the jar.

There were some errors due may be while starting and stopping the watch. This error cannot be eliminated.

...read more.

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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