# 5th Round of 30th year

Series

### 1. ... space snowman(3 points)

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.

\textbf {Bonus:} Generalise the problem for $N\ge$3 spheres. Will the force converge to a finite value for $N~→~∞$ or will it go to infinity?

### 2. ... spheres in viscous fluids(3 points)

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 $C$ is the drag coefficient of the object in the direction of motion, $ρ$ is the density of the fluid, $S$ is the cross-section area and $v$ is 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 $η$ is the dynamic viscosity of the fluid and $r$ is 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?

### 3. ... accurate central collisions(6 points)

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.

### 4. ... on a string(8 points)

Two masses of negligible dimensions and mass $m$ = 100 g are connected by a massless string with rest length $l$_{0} = 1 m and spring constant $k$ = 50 kg·s^{ − 2}. One of the masses is held fixed and the other rotates around it with frequency $f$ = 2 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.

### 5. ... balloon(8 points)

Consider a balloon with mass $m$ (blown up) and volume $V$ filled with helium. An infinite string of length density $τ$ = 10 gm^{ − 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?

### P. ... glasses(8 points)

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.

### E. ... fishing line(12 points)

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.

### S. ... ( points)

statistical physics
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