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Paper boats play a huge role in making our childhood memorable. For some it is an introduction to origami making. For me who is still an origami enthusiast, I still love making paper boats and recently made some for our neighbourhood children who wanted to race them. Due to the unsanitariness of the sewer, a large tub was collected where the race would occur by swirling the tub water with a stick. The undisturbed movements of the boats during swirling of water in the tub and the direction of the movements of the boat when disrupted made me really interest in the motion of fluids. I thought the velocity of the boats at each point would be same but after some research found that this in fact is not true. Rather the velocity of the particle in the fluid, every point in space of flow has a constant velocity vector. Hence, it is extremely important we understand the motion of fluid by studying the motion of particles in the fluid and how the fluid influences the shape of the particles within it since the shape of the paper boats would alter considerably after some time.

With this curiosity I decided to study how the shape and motion of some metal balls would vary by studying their motion in a fluid other than water like glycerin. With this study we would understand some concepts of variation of motion and shape of particles in a fluid.

How does the terminal velocity of the metal ball in cm s^{-1} moving vertically downwards through glycerin depends on the radius (in cm) of the metal balls, determined by measuring the time taken for the ball to fall?

If in a fluid, a particle moves through a definite path and each successive particle then follows the same path having the same velocities while moving through the points of space, then these paths are called ‘lines of motion’ or streamlines. The direction of the velocity of flow at a particular point is given by the tangent at that point on this stream line.

When the velocity of the flow of fluid is too large and the lines of flow change directions too abruptly and becomes disorderly then the flow is said to be turbulent. In this type of motion of the fluid the magnitude and velocity vector at a particular point in space doesn’t remain constant with time and changes continuously. Due to irregular and rotary motion of the fluid, local circular type currents are formed within the fluid called vortices. They can be seen when a solid moves through a fluid.

When the velocity is within a limit, streamline motion occurs. There exists a limiting velocity for respective fluids which when exceeds streamline motion ceases to exist and turbulent motion sets on which is called the critical velocity (*v _{c}*)

The critical velocity depends on the following properties of the fluid (η)

- Co-efficient of viscosity of the fluid (η)
- Density of the fluid (
*ρ*) - Diameter of the tube through which the fluid flows(d)

\(Critical velocity (vc) = R\frac{η}{ρd}\)

Where, R is called the Reynold’s number R and is a constant.

The screw gauge is a U-shaped piece of instrument, where one arm consists of a fixed stud whereas the other is attached to a cylindrical tube. A scale S in centimeter is marked on this cylinder. A screw moves axially when it is rotated by the milled head. The levelled end of the collar is divided into 100 equal divisions forming a circular scale.

The axial displacement for a complete rotation is called the pitch of the screw gauge. The least count of the screw gauge is the axial displacement of the screw for a rotation of one circular division. If there are ‘n’ divisions on the circular scale and ‘m’ divisions of the pitch of the screw then the least count of the screw gauge is given by, Least count \(=\frac{m}{n}\) scale divisions.

The radius of the metal balls measured using a screw gauge is given by -

*r = l + s × least count*

Where l is the linear scale reading and s is the number of circular scale divisions.

**Null Hypothesis:** There is no dependence of the radius of the metal balls on the critical velocity of the fluid.

**Alternate Hypothesis:** There persists a relationship of the radius of the metal balls on the critical velocity of the fluid.

Radius of the metal ball

The diameter of the ball was measured using a screw gauge and the radius was thus determined. The radius has been varied from 0.310 ± 0.001 cm to 0.628 ± 0.001 cm.

The critical velocity of the metal balls moving vertically downward through glycerin.

The time taken for a metal ball to travel a particular length of the tube was measured by a stop-watch. The ball was allowed to fall from a height of 85 cm to 15 cm through a cylinder filled with glycerin. The critical velocity of the ball was determined using the following formula:

\(Terminal velocity = \frac{Distance\ travelled\ by\ the\ ball\ in\ cm}{Time\ taken\ by\ the\ ball\ to\ travel\ in\ s}\,cm s-1\)

Variable | Why it has been controlled? | How is the variable controlled? |
---|---|---|

The length of the tube | If the length of the tube changes then the distance travelled by the metal balls would change which would change their critical velocity | A fixed tube of a constant length is used to measure the critical velocity of the metal balls. The ball was allowed to fall from 85 cm to 15 cm. |

The density of fluid | If the density of fluid changes then the experienced by the ball falling through the fluid changes. Terminal velocity of the metal balls would change for that fluid. | A particular type of fluid e.g, glycerine is used throughout the experiment. |

Least count of the screw gauge | If the least count of the screw gauge changes accuracy in determining the radius of the metal balls used would change. | Same screw gauge is used to measure the radius of the metal balls. |

Temperature of the liquid | Viscosity changes rapidly with temperature which would change the terminal velocity since it depends on the viscosity. | The experiment has been done in an air-conditioned room to maintain a constant temperature. |

Apparatus | Specification | Quantity | Least Count | Uncertainty (±) |
---|---|---|---|---|

Metal Balls | Steel made | 4 | - | - |

Cylindrical tube | Transparent tube of length 1 meter | 1 | - | - |

Screw Gauge | - | 1 | 0.001 cm | 0.001cm |

Stop watch | - | 1 | - | - |

Bottle of Glycerin | - | 1 | - | - |

A metre scale | - | 1 | 0.1cm | 0.001cm |

**Part - A:** Determination of least count of the screw gauge:

**Part - B:** Determination of the diameter of the metal ball:

**Part - C:** Determination of time taken for the ball to fall:

- The cylinder is set vertically on a stand and the chosen experimentally liquid glycerin is poured into the cylindrical tube carefully from the reagent container.
- The height of the liquid column is measured using a meter scale. Two horizontal marks are made on the outer surface of the cylinder to mark our desired lengths at 85.00 ± 0.01 cm and 15.00 ± 0.01 cm.
- The metal ball is thoroughly dipped in glycerin.
- The stop - watch was started when the ball is at the mark of 85.00 ± 0.01 cm.
- The stop - watch was stopped when the ball reaches the mark of 15.00 ± 0.01 cm.
- Repeat steps 1 - 5 for four more times.
- Repeat steps 1 - 6 for all other metal balls.

- The screw gauge is handled under appropriate supervision.
- Gloves are used while pouring the glycerin.
- A spatula is used to drop the balls into the tube.
- The experiment is carried out in a dry room.
- The metal balls are extremely small in radii and hence are handled under supervision and with care.

The experiment is done using the cheapest materials to cut down experiment cost and no wrongful means have been used.

Being environmental conscious, we have used plastic alternatives at every step of the way and the glycerin used is non-lethal.

**Measuring the diameter of the balls using a screw gauge**

No zero - error found

Least count of the circular scale = 0.001 cm

Zero error = - 0.001 cm

Total reading in cm = Main scale reading + Circular scale reading – Zero error

**Sample for 1 ^{st} trial of diameter**

Main scale reading = 6

Circular scale reading = 28

Total reading = 6 × 0.1 cm + 28 0.001 cm = 0.628 cm

Serial number | Diameter of the metal balls (± 0.001 cm) | Radius of the metal balls (± 0.001 cm) | Mean Time of drop (± 0.01 s) | Critical velocity (cm/s) |
---|---|---|---|---|

1 | 0.628 | 0.314 | 2.28 | 30.70 |

2 | 0.472 | 0.236 | 3.25 | 21.53 |

3 | 0.387 | 0.193 | 4.15 | 16.86 |

4 | 0.310 | 0.155 | 5.56 | 12.58 |

**Sample calculation -**

For Row - 1 -

\(Radius of the metal ball = \frac{Diameter\ of\ the\ ball}{2} = \frac{0.628\ ±\ 0.001}{2}= 0.314 ± 0.001 cm\)

** \(Terminal velocity (V) =\frac{Distance\ travelled\ by\ the\ ball\ (d)}{Time\ taken\ for\ the\ ball\ to\ fall t}\)**

Distance travelled by the ball

= 85.00 ± 0.01 cm – 15.00 ± 0.01 cm

= (85.00 -15.00) ± (0.01 + 0.01) cm

= 70.00 ± 0.02 cm

\(Terminal velocity (V) =\frac{Distance\ travelled\ by\ the\ ball\ (d)}{Time\ taken\ for\ the\ ball\ to\ fall t}=\frac{70.00}{2.28} = 30.70 cm s-1\)

**Absolute uncertainty in distance travelled (∆d) = ±0.02 cm**

**Fractional uncertainty in distance travelled \((\frac{∆d}{d})=\frac{±0.01}{2.28}\)**

\(V = \frac{d}{t}\)

\(\frac{∆V}{V}=\frac{∆d}{d}+\frac{∆t}{t}=\frac{± 0.02}{70.00}+\frac{±0.01}{2.28}= ±0.0046\)

**Percentage uncertainty in terminal velocity \(=\frac{∆V}{V}× 100 = ± 0.0046 × 100 = ± 0.46\)**

considered radius of the metal balls along the x-axis and the critical velocity along y-axis since it depends on the radius of the ball. As the radius increases from 0.155 to 0.314 the critical velocity increases from 12.58 to 30.70.

We have also obtained an equation of the trendline: y = 113.97x - 5.1837, where y is critical velocity and x is radius.

Critical velocity of a fluid depends on the co-efficient of viscosity of the fluid which is a characteristic of the fluid and is empirically equal to the tangential viscous force per unit area. Now, if the radius of the metal ball increases, the viscous force per unit area increases and since critical velocity is directly proportional to co-efficient of viscosity , there is a gradual increase in critical velocity of the fluid.

Null Hypothesis:

Critical velocity and radius of the metal balls are not corelated with each other.

Alternate Hypothesis:

Critical velocity and radius of the metal balls are corelated with each other . The equation we obtained from the graph: y = 113.97x + 5.1837 establishes a positive gradient between radius(x-axis) and Critical velocity (y-axis). The value of R^{2} obtained from the graph is 0.9997 . Hence there is 97% co-relation between Critical velocity and radius of the metal balls and they are positively co-related. Hence the null hypothesis has been rejected and alternate hypothesis has been established.

There exists a strong correlation between critical velocity and radius of the metal balls.There is 97% co-relation between Critical velocity and radius of the metal balls and they are positively co-related.

Evaluation:

- The radius of each ball is measured accurately using screw gauge.
- The balls are dropped using a spatula so as to fall at the centre.
- The radii of the balls are measured five times and their mean is taken
- The critical velocity is calculated from the mean time of drop so as to assure minimum error.
- The method is suitable for highly viscous fluid.

Type of error | Source of error | Effect of error | Improvement |
---|---|---|---|

Random | Radii of the metal balls may vary due to temperature. | Variation of critical velocity of the fluid | We have taken five trials and calculated their mean |

Systematic | Heating of the experimental fluid. | Slight variation of actual data | Experiment done in an Air-conditioned room |

Methodological | Backlash error | Variation of radii of the metallic balls | The screw head should be rotated in the same direction. |

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- What Is Streamline in Fluid Mechanics? – SidmartinBio.
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- What Is Screw Gauge and How It Is Used for Measurement?