School of Engineering

And Energy

ENG:670

Measurement and Uncertainty Analysis

Case Study

A crater-formation experiment

Unit Coordinator: David Parlevliet

Submitted By:

Ritu (33255749)

Prabhjot Singh (32872184)

Table of Contents

1. Abstract: 1

2. Introduction: 1

3. Aim: 1

4. Background: 1

5. Apparatus: 2

6. Procedure: 2

7. Methods: 3

8. Calculations: 3

9. Results: 4

10. Analysis: 4

11. Conclusion: 5

12. References 6

1. Abstract:

Crater diameter obtained by sphere of changeable mass thrown in the sand was studied during low speed. The relationship between the Kinetic energy and the crater diameter being obtained same as the crater of planetary meteor.

2. Introduction:

At the point when a ball is dropped into sand, a round cavity is shaped because of the effect of the ball. The span of the cavity differs with the dynamic vitality of the ball at effect. In this exploration, the relation between the crater diameter and kinetic energy will be calculated. The size of crater is identified with the measure of vitality the meteor has as it strikes the ground. The more energy will have long distance or larger the diameter.

This examination will Endeavor to decide the connection between the dynamic Kinetic energy and the ball thrown from various height into sand, and the distance across of the pit framed. 1

It will likewise be resolved whether it is the equivalent as the relationship for Crater.

3. Aim:

The aim of this case study is to create a link between kinetic energy and the Crater diameter and as well as to calculate the n (standard uncertainty) and the coverage interval 2.

Draw a graph for showing the relation in lnD and lnE, by uncertainties in independent variables and the dependent variables. Pass uncertainties up to dependent variable.

Experiment is done at uncertainty of 95%.

All uncertainties being transferred to those variables which are dependents.

Least square methods will be used just to calculate the exact values of variables.

4. Background:

A Crater is shaped when a quick moving ball strikes the surface of, for instance, a strong planet 1. By concentrate the connection between the measurement of the pit what’s more, the kinetic energy of the affecting item, it is conceivable to find which energy disseminating system commands (as models, vitality might be dispersed by deformation of material, discharge of material from the cavity and the making of seismic waves) 1 3.

On the off chance that the overwhelming procedure by which KE is scattered is plastic twisting,

at that point it is anticipated that the Crater diameter, D,

D = cE^1/.3. (1)

By difference, if most of the occurrence energy is exchanged to sand which is launched out from the creator, at that point the cavity distance across is anticipated to be identified with the occurrence of KE by the condition,

D = cE^1/4 (2)

In equation (1) and (2), c is a consistent.

5. Apparatus:

The apparatus used for this experiment contains various ball with different masses, sand in a container, a large and small ruler, a balance scale, Vernier calliper to measure the diameter of ball, weight machine.

Fig. (1) Main diagram of Apparatus used.

6. Procedure:

• First, measure the ball’s mass by the weight machine, ball may have any type such as plastic, steel, etc. A plastic cap weight is being measured before calculating the weight of the ball properly 2.

• Put the sand in the container than make the level of that sand with the help of any scale, make sure the sand should be properly levelled.

• Use the large scale to measure the height of the ball, in our experiment this large scale is also being used as stand.

• Properly shake the container of sand back and forth for level of the sand.

• Do same procedure again and again at given height, the diameter we measured is an indication for the diameter measurement uncertainty.

• By using the bearing of other ball, drop every ball from various heights and measure the diameter of the Crater every time.

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Fig.2 Sand Container Fig.3 Weight Machine

7. Methods:

Masses of balls 66.78 g,5.31 g, 9.26 g,13.75 g were thrown from the height in between 25 cm to 150 cm in the container which is full of sand. As per the height various Craters’ diameters being formed and KE is being calculated with the help of height and masses of the balls 1 .The sand was spread evenly to a depth of 10 cm. A small lamp was used to. The Crater diameter, D, was measured by the small scale where the minimum distance is of 1mm. After the measurement of the diameter of Crater sand being levelled again for another balls. Various Craters’ diameter being measured for the same ball at same height and after that average of those has been taken.

Diameter of ball Mass(g) Height(cm) Kinetic energy, E(J) Crater Diameter Ln (K.E) Ln (D)

2.5 66.78 80 0.52356 8,8.4,7.8,8.2 -0.64 -2.513

2.5 66.78 33 0.21597 7,6.5,7.3,6.8 -1.5326 -2.6736

1.5 13.75 100 0.13475 5.4,5.2,6,5.8 -2.0043 -2.882

1.2 9.26 63 0.05718 4.8.5.2,4.9,5 -2.8615 -3.0007

1.1 5.61 53 0.02913 4.5,4.3,4.2,4 -3.5356 -3.158

Table 1.: Crater diameter values for kinetic energy (K.E) values.

8. Calculations:

Here for the calculation we considered all the values such as mass, height of the ball, acceleration, crater diameter. Multiples values are here for the diameter of crater but during our measurement we took all values in different units so as we know kinetic energy always comes in joule so for that one mass should be in kg and height and diameters should be in meters.

Mass = 66.78 g and 0.06678 kg

Height = 33 cm and 0.33 m

Diameter= 6.9 cm and 0.069 m

And g= 9.8 m/s^2

Then, LnD= -2.513

Kinetic energy will be

KE= mgh

= 0.06678*0.33*9.8

=0.21597 J

LnE= -0.64

Same procedure will be followed by other values.

9. Results:

Table 1 contains various values of the ball for the Kinetic energy and the diameter of the Crater. All these values being calculated based upon the height and the mass of the balls which is then used as to calculate the kinetic energy by using mass, acceleration and the height.

E = mgh …… (3)

M= mass of the ball, g = acceleration due to gravity (9.8 m/s^2).

1. Uncertainty for the average diameter

2. Ball’s height at the sand’s surface.

3. KE of ball and its uncertainties.

10. Analysis:

The connection between the Kinetic energy and the Crater diameter, D, of the ball can also be written in form of may be written D = cEn.

Least square method is being used to calculate the data in for the given table.

By this technique various assumption being assumed.

1. The errors are restricted to those variables which are dependent, such as D, diameter.

2. The scatters data size of the line neither should decrease nor increases by the predictor variable range 4.

For the verification of validity of assumption (1) for this case study fractional uncertainty of D and E being compared.

Uncertainty in E

It depends upon the mass, acceleration and the height of the ball.

Here, some uncertainties are negligible such as internal energy because of the temperature and the air changes 1.

• Since the balance for the resolution ?, is ? = 0.01 g, in the ball’s mass standard uncertainty is, ?/?12 = 0.002 89 g.

Uncertainty in D,

• Four values of D have been used here for every ball at each height here we can use Type A uncertainty for uncertainty of every energy.

• Linear square methods being used here for the calculation of all uncertainty’s parameters such as D, and KE.

• As the fractional uncertainty in D is consistently greater than that in E, we proceed to analyse the data using unweighted linear least-squares in which we assume that error is confined to the dependent variable, D.

Fig.4 Variation in KE with the crater diameter in form of log scale.

11. Conclusion:

Our observation saying ‘the higher from which the ball was dropped the greater the diameter of Crater would be’ was right since when the ball is falling it develops more speed because of increasing speed. The more speed it develops the more power is being utilized quickening it which is then applied on the sand. The more power applied onto the sand the more sand it can push away along these lines making a large hole. Our other assumption saying ‘after a specific range from which the ball was being dropped the territory of the cavity would remain the equivalent’ was right since when a free-falling item first sets off it has considerably more power quickening it than obstruction backing it off. As the speed rise the resistance develops.

12. References

1 “Formation of Craters in Sand,” ISB Journal of Physics, vol. 1, no. 1, 2007.

2 L. Kirkup , and R. B. Frenkel, An Introduction to Uncertainty in Measurement : Using the GUM (Guide to the Expression of Uncertainty in Measurement), Cambridge University Press, 2006.

3 Amrozia Shaheen, Asma Khalid and Muhammad Sabieh Anwar, “Craters in Sand,” LUMS School of Science and Engineering, Perth, 2015.