Physics Olympiad (English Version)

1. Calculate the period of small oscillations of a hydrometer (Figure.1) which was slightly pushed down in the vertical direction. The mass of the hydrometer is m = 50 g, the radius of its tube is r = 3.2 mm, the density of the liquid is p = 1.00 g/cm³. The resistance of the liquid is assumed to be negligible.








Answer: 2,477 s

2. Find the frequency of small oscillations of the arrangement illustrated in Fig. 2. The radius of the pulley is R, its moment of inertia relative to the rotation axis is I, the mass of the body is m, and the spring stiffness is x. The mass of the thread and the spring is negligible, the thread does not slide over the pulley, there is no friction in the axis of the pulley.














3. A vessel contains a mixture of nitrogen (m₁ = 7.0 g) and carbon dioxide (m₂ = 11 g) at a temperature T = 290 K and pressure p_o, = 1.0 atm. Find the density of this mixture, assuming the gases to be ideal.
Answer: 1,51 g/l

4. A spool with thread wound on it, of mass m, rests on a rough horizontal surface. Its moment of inertia relative to its own axis is equal to I = γmR², where y is a numerical factor, and R is the outside radius of the spool. The radius of the wound thread layer is equal to r. The spool is pulled without sliding by the thread with a constant force F directed at an angle a to the horizontal (Fig. 3). Find:
(a) the projection of the acceleration vector of the spool axis on the x-axis;
(b) the work performed by the force F during the first t seconds after the beginning of motion.














5. A uniform cylinder of radius R is spinned about its axis to the angular velocity ω_o and then placed into a corner (Fig. 4). The coefficient of friction between the corner walls and the cylinder is equal to k. How many turns will the cylinder accomplish before it stops?

1. Why is the following situation impossible? You are in the high-speed package delivery business. Your competitor in the next building gains the right-of-way tobuild an evacuated tunnel just above the ground all the way around the Earth. By firing packages into this tunnel at just the right speed, your competitor is able to send the packages into orbit around the Earth in this tunnel so that they arrive on the exact opposite side of the Earth in a very short time interval. You come up with a competing idea. Figuring that the distance through the Earth is shorter than the distance around the Earth, you obtain permits to build an evacuated tunnel through the center of the Earth (Fig.
P15.86). By simply dropping packages into this tunnel, they fall downward and arrive at the other end of your tunnel, which is in a building right next to the other end of your competitor’s tunnel. Because your packages arrive on the other side of the Earth in a shorter time interval, you win the competition and your business flourishes. Note: An object at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r (the reddish region in Fig. P15.86). Assume the Earth has uniform density.














2. A smaller disk of radius r and mass m is attached rigidly to the face of a second larger disk of radius R and mass M as shown in Figure P15.84. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated
through a small angle θ from its equilibrium position and released.
(a) Show that the speed of the center of the small disk as it passes through the equilibrium position is




(b) Show that the period of the motion is






3. Determine the period of oscillations of mercury of mass m = 200 g poured into a bent tube (Fig. 4.5) whose right arm forms an angle θ = 30° with the vertical. The cross-sectional area of the tube is S = 0.50 cm². The viscosity of mercury is to be neglected.









Answer: 0,788 s

4. A smooth vertical tube having two different sections is open from both ends and equipped with two pistons of different areas (Fig. 2.1). Each piston slides within a respective tube section. One mole of ideal gas is enclosed between the pistons tied with a non-stretchable thread. The crosssectional
area of the upper piston is ΔS = 10 cm² greater than that of the lower one. The combined mass of the two pistons is equal to m = 5.0 kg. The outside air pressure is Po = 1.0 atm. By how many kelvins must the gas between the pistons be heated to shift the pistons through 1 = 5.0 cm?










Answer: 0,91 K

5. A block is placed on a ramp of parabolic shape given by the equation y = x²/20, Figure 1. If μs = 0.5, what is the maximum height above the ground at which the block can be placed without slipping?







(Figure 1)
Answer: 1,25 m

(Figure 2)

6. A small cube placed on the inside of a funnel rotates about a vertical axis at a constant rate of f rev/s. The wall of the funnel makes an angle θ with the horizontal (Figure). If the coefficient of static friction is μ and the centre of the cube is at a distance r from the axis of rotation, show that the largest frequency for which the block will not move with respect to the funnel is




7.In prob. (6), show that the minimum frequency for which the block will not move with respect to the funnel will be




8.Consider the rigid plane object of weight Mg shown in Fig. 2 pivoted about a point at a distance D from its centre of mass and displaced from equilibrium by a small angle ϕ. Such a system is called a physical pendulum. Show that the oscillatory motion of the object is simple harmonic with a period given by
where I is the moment of inertia about the pivot point.








(Figure 3)




9. A man standing in front of mountain at a certain distance beats a drum at regular intervals. The drumming rate is gradually increased and he finds the echo is not heard distinctly when the rate becomes 40/min. He then moves nearer to the mountain by 90m and finds that the echo is again not heard when the drumming rate becomes 60/min. Calculate:
 (a) the distance between the mountain and the initial position of the man and the mountain and
 (b) the velocity of sound.
Answer: (a) x = 270 m
               (b) v = 360 m/s

10.Water flows in a horizontal pipe of varying cross-section. Two manometer tubes fixed on the pipe, Fig. 3, at sections A1 and A2 indicate a difference Δh in the water columns. Calculate the rate of flow of water in the pipe.

1. As shown in Figure 1, two tanks of different cross-sectional areas are connected by a pipe. The tanks are filled with water, and openings A and B are tightly sealed with pistons that can move without friction. Initially, both pistons are in equilibrium at the same height. The cross-sectional area of B is n times larger than that of A.
As shown in Figure 2, a force of magnitude F is applied vertically downward to the piston at A, causing it to move downward by distance h. As a result, the piston at B moves upward by distance h'. Next, a force of magnitude F' is applied vertically downward to the piston at B. As a result, both pistons return to their original positions shown in Figure 1. From 1-9 below choose the combination that correctly expresses h' and F'.


Answer: 5

2. Block A have weight 100 N tied with ropes horizontally in C (see figure). Block B have weight 500 N. Coefficient of friction between A and B = 0.2 and the coefficient of friction between the floor and B = 0.5. The magnitude of the force F at least to shift the beam B is. ..

Answer: 320 N

7. A projectile of mass m is shot (at velocity v) at a target of mass M, with a hole containing a spring of constant k. The target is initially at rest and can slide without friction on a horizontal surface. Find the distance  ∆x that the spring compresses at maximum.