Series 5, Year 30

Consider a snowman consisting of 3 homogeneous spheres of density $ρ$ with centres on a line, floating in free space. The smallest sphere (the head) has radius $r$ and each consecutive sphere has twice the radius of the previous one. Our snowman is the only thing in the universe and it does not rotate in any way. Find the force holding the head to the rest of the snowman.

Bonus: Generalise the problem for $N\ge3$ spheres. Will the force converge to a finite value for $N→∞$ or will it go to infinity?

When solving problems involving drag in air or in general a fluid, we use Newton's resistance equation

$$F=\frac{1}{2}C\rho Sv^2\,,$$

where $Cis$ the drag coefficient of the object in the direction of motion, $ρis$ the density of the fluid, $Sis$ the cross-section area and $vis$ the velocity of the object. This is usually quite accurate for turbulent flow. We are interested in a sphere, for which $C=0.50$. In the case of laminar flow, we usually use Stokes' law

$$F = 6 \pi \eta r v\,,$$

where $ηis$ the dynamic viscosity of the fluid and $ris$ the radius of the sphere. Is there a velocity for which the two resistance forces are equal for the same sphere?. How will this velocity depend on the radius of the sphere?

Consider 3 equal non-rotating discs moving in a straight line in the order 1, 2, 3 without friction or any other resistance forces on a horizontal surface. Discs 1 and 2 are moving to the right and disc 3 is moving against them to the left. We know that the velocity of disc 1 is larger than that of disc 2. How do the final velocities (after all collisions) depend on the order in which the collisions take place? What are these velocities? (Do not forget that all answers must be properly justified). Bonus: Discs have different masses.

Two masses of negligible dimensions and mass $m=100g$ are connected by a massless string with rest length $l_{0}=1\;\mathrm{m}$ and spring constant $k=50\;\mathrm{kg}\cdot \mathrm{s}^{-2}$. One of the masses is held fixed and the other rotates around it with frequency $f=2\;\mathrm{Hz}$. The first mass can rotate freely around its axis. At one point the fixed mass is released. Find the minimal separation of the two masses during the resulting motion. Do not consider the effects of gravity and assume the validity of Hook's law.

Consider a balloon with mass $m$ (blown up) and volume $V$ filled with helium. An infinite string of length density $τ=10gm^{-1}$ is tied to the balloon. Assuming the atmosphere is isothermal, in which the pressure depends on height $z$ as

$p=p_0e^{-z/z_0}\$,,

($z_{0}$ is a parameter of the atmosphere), what is the maximum height the balloon will reach?

Describe the imaging system of a microscope (consisting of two convex lenses) and that of a Keplerian telescope. Explain the difference in function and construction of a microscope and a telescope and sketch the rays passing through the systems. How can we usefully define magnification for these optical systems? Derive the equations for magnification.

Measure the shear modulus $G$ (modulus in torsion) of a fishing line. Unfortunately, we are unable to mail the fishing line samples abroad, we therefore ask that you obtain one by yourself and include pictures of the line (and the reel it came from) you use in your solution.

1. Try to describe in your own words how and for what purpose linear regression is used (it is sufficient to briefly describe the following: two significant applications of linear regression, least squares method, maximum likelihood estimation, linear regression model, basic graphical methods of regression diagnostics). It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
2. In the attached data file linreg1.csv you may find the results of a certain physical experiment, in which we measured the pairs of data $(x_i, y_i)$. We want to fit the measured data with a theoretical function in this case a parabola in the form \[\begin{equation*} f(x) = ax^2 + bx + c . \end {equation*}\] Determine the value of the coefficient $a$ and its uncertainty. It is not necessary to use regression diagnostics.
3. In the attached data file linreg2.csv you may find the results of a certain physical experiment, in which we measured the pairs of data $(x_i, y_i)$. We want to fit the measured data with a theoretical function, in this case a logarithmic function in the form \[\begin{equation*} f(x) = a+ b \cdot \log (x)  . \end {equation*}\] Plot the measured data into a graph with the fitting function and briefly comment on it. It is not necessary to use regression diagnostics.
4. Suppose we have measured pairs of data $(x_i, y_i)$ and want to fit them with a linear function in the form \[\begin{equation*} f(x) = a+ bx  . \end {equation*}\] Derive the exact formula for calculating the regression parameters. You may use any and all sources, if you cite them correctly. (Actually derive the formula, do not just write it.)

Bonus: In the tasks b) and c) perform regression diagnostics and discuss, whether all necessary criteria (assumptions) are met.

For data processing and creating the plots, you may use the R programming language. Most of these tasks can be solved by slightly altering the attached scripts.

• script in the UTF-8 coding
• script in the CP-1250 coding
• vzor.csv
• linreg1.csv
• linreg2.csv