Good to have. o Teacher(s) o Scientific Method. o High-speed camera o Tracker (

Transcription

1 30 th IYPT

2 Good to have o Teacher(s) o Scientific Method o High-speed camera o Tracker ( o MATHEMATICA o PYTHON ( 2

3 Problem 1 : Invent yourself (Category B) Construct a passive device that will provide safe landing for an uncooked hen s egg when dropped onto a hard surface from a fixed height of 2.5 m. The device must fall together with the egg. What is the smallest size of the device you can achieve? 3

4 Impulse-momentum theorem Assuming air resistance is negligible, and we have an egg freely falling from rest: 2 2. is the linear momentum of the egg after falling a height of. To bring this egg to rest in time Δ requires an average force given by Δ 2. Assuming 50 g, 9.8 m s,thenwith2.5 m, N s. Δ s N

5 Air resistance According to Newton s 2 nd law,. So, with 2.5 m: 1exp, 5 2 It follows that when 0, And, 5 2 1exp. ProductLog exp ProductLog exp

6 Air resistance (cont.) Assuming 50 g, 9.8 m s, then: Indeed, with 0.7, s,and0.035ns. 6

7 Air resistance (cont.) According to Newton s 2 nd law, So, with 2.5 m:. tanh, 5 2 ln cosh. It follows that when 0, cosh exp 5 2. And, 1exp 5 2 tanh

8 Air resistance (cont.) Assuming 50 g, 9.8 m s, then: Indeed, with 1.0, s,and0.035ns. 8

9 Problem 2 : Balloon airhorn (Category B) A simple airhorn can be constructed by stretching a balloon over the opening of a small container or cup with a tube through the other end (see Figure). Blowing through a small hole in the side of the container can produce a sound. Investigate how relevant parameters affect the sound (frequency spectrum, intensity, etc.). o o 9

10 Problem 3: Single lens telescope (Selection) A telescope can be built using a single lens, provided that a small aperture is used instead of an eyepiece. How do the parameters of the lens and the hole influence the image (e.g., magnification, sharpness, and brightness)? Questions. o How does a Keplerian refracting telescope work? o How does a pinhole camera work? 10

11 A two-lens system and angular magnification Consider the ratio tan tan. In a telescope, tan, tan,,,and. So, its angular magnification:. 11

12 Problem 8 : Visualising density (Category B) Schlieren photography is often used to visualise density variation in a gas. Build a Schlieren setup and investigate how well it can resolve density differences. o o

13 Problem 9 : Ball in a tube (Category B) A sealed transparent tube is filled with a liquid and contains a small ball. The tube is inclined and its lower end is attached to a motor such that the tube traces a conical surface. Investigate the motion of the ball as a function of relevant parameters. Some things to vary:,, density, viscosity, size of ball, etc. 30

14 Origin of buoyant force where is pressure and is its gradient. Here, and refer to the volume of the ball and the boundary of respectively. The above is a result of:. Here, is an arbitrary scalar field and is a constant vector, and we have used the divergence theorem and product rule of differentiation. For the spherical ball, cos sin sin cos sin sin cos 2 cos cos 2 3 cos

15 Origin of buoyant force (cont.) In this case,, 4 3. cos 32

16 Rotating frames Let be an inertial frame and let be a frame rotating about the axis with angular velocity. Call the basis vectors of :,,,andthoseof :,,. Consider a particle at rest in. From the perspective of, its velocity is. This formula clearly applies to,, :. Now, consider a general time-dependent vector :. 33

17 Rotating frames (cont.) It follows that in general: In particular,,. The difference in velocity measured in the two frames is the relative velocity of the frames, which depends on position. And,

18 Rotating frames (cont.) Since is an inertial frame, Newton s 2 nd law is 2. So, in the rotating frame : 2. These fictitious forces affect all bodies equally, like gravity, since all contain factor of. 35

19 Problem 10 : Pulling glasses apart (Category B) Put a thin layer of water between two sheets of glass and try to separate them. Investigate the parameters affecting the required force. 36

20 Capillary bridge and capillary-bridge force Two wetted surfaces can stick together with great strength if the equilibrium contact angle 2. The pressure difference between the inside and the outside of a curved surface that forms the between air and water, or Laplace pressure Δ 1 2cos The pressure difference is caused by the surface tension of the interface between air and water. 37

21 Problem 11 : Hair hygrometer (Category B) Asimplehygrometer can be built using human hair. response time as a function of relevant parameters. Investigate its accuracy and Human hair stretches in moist air and contracts in dry air. o 38

22 Relative humidity and hygrometer o The relative humidity of an air-water mixture is defined as the ratio of the partial pressure of water vapour ( ) in the mixture to the equilibrium vapour pressure of water vapour ( ) over a flat surface of pure water at a given temperature:. o There are several empirical formulas that can be used to estimate the equilibrium vapour pressure of water vapour as a function of temperature in C: Antonine equation, Goff-Gratch equation, Magnus-Tetens approximation, etc. For example, the Buck formula: exp

23 Problem 13: Resonating glass (Category B) A wine glass partially filled with liquid will resonate when exposed to the sound from a loudspeaker. Investigate how the phenomenon depends on various parameters. o Some things to vary: o wine glasses geometry, elastic property, o liquid density, viscosity, amount, etc. 46

24 Problem 17 : Vacuum bzooka (Category B) A vacuum bazooka can be built with a simple plastic pipe, a light projectile, and a vacuum cleaner. Build such a device and maximise the muzzle velocity. (a) Tube is evacuated with end seals in place. (b) Seal is punctured and air rushes into vacuum. (c) Pressure differential accelerates ball and some air leaks past. (d) Air ahead of the ball is compressed and breaks muzzle seal. (e) Ball exits through open muzzle. o 59

25 Zeroth order analysis of the muzzle velocity The acceleration of the projectile, mass, would be a constant:, where and are the atmospheric pressure and projectile cross-sectional area, respectively. It follows that maximum muzzle velocity is dependent on the tube length :

26 First order analysis of the muzzle velocity Assuming that the projectile fits and seals the barrel perfectly, and that effects due to compressibility, friction, and viscosity are negligible, then Newton s 2 nd law:, where is the air density and is the position of the projectile (which begins at 0, 0). It follows that 1 1 1, where characteristic length.so,

27 First order analysis of the muzzle velocity (cont.) It follows that 1 1 The maximum muzzle velocity,

28 How does one measure muzzle velocity? A ballistic pendulum. 1 2 By conservation of total linear momentum,. By conservation of total mechanical energy,