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Problems in Physics, April 2017

Please read the rules of the competition.


Problems with sign 'G'

Deadline expired on 10 May 2017.

G. 599. The acceleration of an object starting from rest at \(\displaystyle t_0=0\), and moving along a straight path, is \(\displaystyle 4~{\rm m/s}^2\). The magnitude of the acceleration of another object, undergoing uniform circular motion of radius 9 m, is also \(\displaystyle 4~{\rm m/s}^2\).

\(\displaystyle a)\) What is the instantaneous speed of each object at \(\displaystyle t_1=5\) s?

\(\displaystyle b)\) How much distance do they cover during the 5 s?

(3 points)

This problem is for grade 1 - 9 students only.

Solution (in Hungarian)

G. 600. A helicopter flies between two cities daily. Its normal speed in each direction is 120 km/h. On a windy day in one direction its speed was 140 km/h, whilst in the other direction it was 100 km/h, thus the total time of the whole journey increased by 15 minutes.

What is the distance between the two cities?

(3 points)

This problem is for grade 1 - 9 students only.

Solution (in Hungarian)

G. 601. A 100 gram and a 50 gram weights were glued together, and hung to the end of a vertical spring. When the glue dried out the 50 gram weight came unstuck, and fell down.

At what acceleration did the other object start to move?

(3 points)

This problem is for grade 1 - 9 students only.

Solution (in Hungarian)


Problems with sign 'M'

Deadline expired on 10 May 2017.

M. 368. Let us experiment with a spinning top, or with any other type of gyroscope. Measure how the number of revolution of the top changes as a function of time, once it was set in motion and left to spin.

(6 points)

Statistics


Problems with sign 'P'

Deadline expired on 10 May 2017.

P. 4927. When a battery of unknown internal resistance and electromotive force is loaded with a resistor of resistance 10 ohm the terminal voltage across the battery is 6 V and the energy stored in the battery decreases by 7.2 J in each second.

\(\displaystyle a)\) What is the internal resistance of the battery?

\(\displaystyle b)\) What is its electromotive force?

\(\displaystyle c)\) What is the terminal voltage when the resistance of the load is increased to 20 ohms?

(4 points)

Solution (in Hungarian)

P. 4928. \(\displaystyle a)\) At what accelerations do the trolleys shown in the figure start to move if the wheels can roll easily, and the mass of the pulley and air drag are negligible? Data: \(\displaystyle m_1=1\) kg, \(\displaystyle m_2=2\) kg, \(\displaystyle M=5\) kg.

\(\displaystyle b)\) For other mass values, in what interval can the value of the initial acceleration of the trolley of mass \(\displaystyle M\) be?

(5 points)

Solution (in Hungarian)

P. 4929. A rigid rod of length \(\displaystyle 3L\), and of negligible mass, shown in the figure can rotate in the vertical plane about a fixed horizontal axle, which is at a distance of \(\displaystyle L\) from the left end of the rod. To the ends of the rod small objects of masses \(\displaystyle m\) and \(\displaystyle 2m\) are attached, and then at a certain moment the rod is released from its horizontal position.

\(\displaystyle a)\) Determine the speeds of the objects when the rod is vertical.

\(\displaystyle b)\) At this moment what is the force exerted by the rod on the axle?

\(\displaystyle c)\) What is the acceleration of the objects at the moment right after the rod was released?

(4 points)

Solution (in Hungarian)

P. 4930. There is a uniform density right-triangular cross-section prism of mass \(\displaystyle M\) on the horizontal ground. The prism lies on its smallest face, and the surface of the ground is rough enough. The acute angle of the triangle at its vertex which is on the ground is \(\displaystyle \alpha=60^\circ\). A thread attached to the midpoint of the top edge of the prism, and a small object of mass \(\displaystyle m\) is fixed to the other end of the thread. The bob of the simple pendulum is released from the horizontal position as shown in the figure. The prism is observed to tip when the thread makes an angle of \(\displaystyle 20^\circ\) with the horizontal.

\(\displaystyle a)\) What is the mass ratio \(\displaystyle M/m\)?

\(\displaystyle b)\) To what value should this ratio \(\displaystyle M/m\) be increased in order that the moving pendulum should not be able to make the prism tip at all?

(5 points)

Solution (in Hungarian)

P. 4931. In a thermally insulated horizontal cylinder an easily moveable heat conducting piston separates two samples of gas of the same type, and of different temperature and different pressure. Initially the piston is fixed, but later the fixing ceased. What initial conditions may result in that after the equilibrium is reached, the piston is not on that side where initially the gas had smaller pressure?

(4 points)

Solution (in Hungarian)

P. 4932. The figure shows the cyclical processes of a heat engine which is operated with a certain amount of ideal, noble gas. Both of the cyclical processes \(\displaystyle ABCA\) and \(\displaystyle CDEC\) consist of isothermal isobaric and isochoric processes. What is the ratio of the two efficiencies?

(5 points)

Solution (in Hungarian)

P. 4933. Two unfixed spheres of mass 1 g, and of radius 4 mm are charged to \(\displaystyle Q_1=10^{-10}\) C, and \(\displaystyle Q_2=-2\cdot 10^{-10}\) C and are placed at a distance of 1 m. The spheres are left to move freely.

How long does it take for them to be at a distance of 25 cm?

(5 points)

Solution (in Hungarian)

P. 4934. A 2-metre long straight current carrying wire radiates energy to its environment at a rate of 10 J per second, whilst its temperature remains the same.

\(\displaystyle a)\) What is the electric field strength in the wire, if the diameter of the wire is 4 mm, and the magnitude of the magnetic flux density at the surface of the wire is 0.002 Vs/m\(\displaystyle {}^2\)?

\(\displaystyle b)\) What type of material may the wire be made of?

(4 points)

Solution (in Hungarian)

P. 4935. The wavelengths of a photon and an electron are the same. Which of them has greater kinetic energy?

(5 points)

Solution (in Hungarian)

P. 4936. During the decay of a stationary, free neutron what may the greatest value of the kinetic energy of the electron be?

(5 points)

Solution (in Hungarian)

P. 4937. In the future men will be able to build spaceships which are able to take us to distant star systems. Suppose that one of such a spaceship leaves the Earth at the escape speed, and due to its special engine every day its kinetic energy is doubled. (Its rest mass remains the same.)

Estimate how much does the captain of the spaceship get older when he travels to Alpha Centauri, which is at a distance of 4.3 light years from the Earth.

(6 points)

Solution (in Hungarian)


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