KöMaL Problems in Physics, December 2024
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Problems with sign 'M'Deadline expired on January 15, 2025. |
M. 436. Investigate the bounces of a ping-pong ball on at least three different solid surfaces (wood, glass, paving slabs, etc.) using a sound processing program or a phone application. Measure the coefficient of restitution, typical for the bounces.
(6 pont)
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Problems with sign 'G'Deadline expired on January 15, 2025. |
G. 869. A car's front and rear wheels are at the vertices of a rectangle with sides 4 m and 2 m, as shown in the figure.
a) If the centre of the line segment between the rear wheels turns around a circle of radius \(\displaystyle R=10~\mathrm{m}\) when the car turns, what is the radius of the circles drawn by the wet wheels of the car on the dry asphalt?
b) During turning, what is the angle turned by the front wheels about the vertical axis?
(3 pont)
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G. 870. Calculate the force needed to keep a standard ping-pong ball under water.
(3 pont)
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G. 871. The temperature of the coolant of a car is \(\displaystyle 240~^\circ\mathrm{F}\) when it leaves the engine of the car. After it passes through the car's radiator, its temperature drops to \(\displaystyle 175~^\circ\mathrm{F}\). Calculate how much heat is released from the engine through the cooling system to the environment in 1 hour if there are 2 gallons of coolant in the car, which circulates through the cooling system in 15 seconds. The coolant has a specific heat of \(\displaystyle 3.5~\tfrac{\mathrm{J}}{\mathrm{g}\,^\circ\mathrm{C}}\) and density equal to that of water.
(3 pont)
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G. 872. A lamp, of resistance \(\displaystyle 100~\Omega\), a toaster of resistance \(\displaystyle 50~\Omega\) and an electric water filter of resistance \(\displaystyle 500~\Omega\) are connected in parallel to the \(\displaystyle 230~\mathrm{V}\) voltage supply. What is the resistance of the electric iron that draws as much current from the mains supply as the three appliances together, and what is this current?
(3 pont)
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Problems with sign 'P'Deadline expired on January 15, 2025. |
P. 5607. A small body is projected horizontally from the top of a fixed slope, and hits the bottom of the slope (see the figure). On impact, its velocity makes an angle of \(\displaystyle \beta=19^\circ\) with the plane of the slope. What is the angle of inclination of the slope?
(3 pont)
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P. 5608. An explosive projectile of mass \(\displaystyle 8~\mathrm{kg}\) is fired at an angle of \(\displaystyle 60^\circ\) with respect to the horizontal, with an initial speed of \(\displaystyle 120~\mathrm{m}/\mathrm{s}\). It explodes at the top of its trajectory into two pieces of masses \(\displaystyle 3~\mathrm{kg}\) and \(\displaystyle 5~\mathrm{kg}\) such that the relative velocities of the two pieces are perpendicular to the plane of the trajectory of the projectile. 80% of the \(\displaystyle 12~\mathrm{kJ}\) energy released in the explosion is used to increase the kinetic energy of the pieces. From the position of the launch
a) how far and
b) at what speed will the two pieces reach the ground?
(Neglect air resistance.)
(4 pont)
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P. 5609. Approximate a beverage can of volume \(\displaystyle 330~\mathrm{ml}\) with a uniform mass distribution cylinder of height \(\displaystyle H=14.6~\mathrm{cm}\) and inside diameter of \(\displaystyle d=5.4~\mathrm{cm}\). The mass of the can is \(\displaystyle M=14~\mathrm{g}\). How much water should be poured into the can in order that the centre of mass of the system be as low as possible? At what height is the centre of mass then?
(4 pont)
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P. 5610. The minor planet Eros approaches the Sun at \(\displaystyle 1.13~\mathrm{AU}\) (when it is at the closest point), and it is \(\displaystyle 1.78~\mathrm{AU}\) away from the Sun at the furthest point of its orbit. AU is the abbreviation of astronomical unit, 1 AU is the mean distance between the Sun and the Earth. What is the maximum and minimum speed of the minor planet Eros?
(5 pont)
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P. 5611. The ``double yo-yo'' shown in the figure consists of two identical discs with uniform mass distribution and the yarns wound on them.
The two discs are released with zero initial velocity such that the yarns are vertical.
a) Which of the two discs' axis will have a greater speed after a certain time elapsed, and how many times this speed is greater than the speed of the axis of the other disc?
b) Which disc's angular speed will be greater after a certain time elapsed, and by what factor will this angular speed be greater than that of the other disc?
(At the moment in question, the yarns have not yet been unwound from the discs.)
(5 pont)
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P. 5612. A \(\displaystyle 1000~\mathrm{J}\) thermal energy is transferred to a sample of nitrogen gas, which expands at a constant temperature of \(\displaystyle 25~^\circ\mathrm{C}\). Then the gas further expands adiabatically such that the gas cools to a temperature of \(\displaystyle 0~^\circ\mathrm{C}\). The same final state could have been reached, if the gas had first expanded adiabatically and then isothermally. In this case, how much heat would have to be transferred to the gas during the isothermal process?
(4 pont)
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P. 5613. A flat disc, made of 1 mole copper, is moved at a horizontal velocity of \(\displaystyle 1~\mathrm{m}/\mathrm{s}\) in a horizontal and uniform magnetic field of \(\displaystyle 1~\mathrm{T}\), which is perpendicular to the velocity of the copper disc. The disc is positioned so that both its base and top are horizontal. The diameter of the base and top plates is twenty times the height of the disc. Estimate the number of electrons that accumulate on the negatively charged side of the disc due to the motional induction.
(4 pont)
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P. 5614. There are several radioactive isotopes that can both undergo positive beta decay and electron capture (both of which produce the same nuclide). Which has a higher decay energy and by how much?
(4 pont)
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P. 5615. Using an ideal battery with an electromotive force of \(\displaystyle U_0=4~\mathrm{V}\), a load with a resistance of \(\displaystyle R=0.5~\Omega\), a switch and two düristors (D), we construct the circuit shown in figure a. A düristor is a circuit element consisting of an ideal coil with inductance \(\displaystyle L=1~\mathrm{H}\) and a non-linear resistor of resistance \(\displaystyle r\), which are connected in series. The \(\displaystyle U_r(I_\mathrm{D})\) characteristics of the resistor is shown in figure b. a) In the stationary (that is, constant in time) case, what are the currents in the sub-branches?
Hint: See problem P. 5604. and its solution in the workbook.
b) At the moment \(\displaystyle t=0\) we turn on the switch. Sketch the currents flowing through the düristors as a function of time. What will these currents be after a long time elapsed?
Hint: Suppose that during this time, due to symmetry, the same current flows through each düristor.
c) During the equilibrium state a small imbalance occurs: the current through one of the düristors decreases by a very small amount, and the current of the other increases by the same amount. Plot the currents of the two düristors as a function of time, after this imbalance occurs. What will the currents of each of the two düristors be after a long time elapsed? What can we say about the stability of the stationary solutions given in a)?
Hint: We can use that if the small imbalance is symmetric with respect to the first equilibrium state, then the sum of the currents through the two düristors remains constant.
(6 pont)
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