How does the Ripple factor of an AC (alternating current) to DC (Direct current) bridge rectifier circuit depend on the permittivity of the material placed between the two parallel plates of the capacitor filter, within the permittivity range of 1.0 to 9.0?

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:

 

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

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

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.