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Problems in Physics, November 2015

Please read the rules of the competition.


Problems with sign 'M'

Deadline expired on 10 December 2015.

M. 354. Attach a spindle perpendicularly to a (preferably uniform density) rod of length \(\displaystyle h\) at a distance of \(\displaystyle \frac{1}{10}h\) from the centre of the rod. Hang the spindle horizontally with threads of lengths \(\displaystyle L\) (bifilar suspension), then make the rod swing perpendicularly to the plane determined by the threads (as shown in the figure).

Investigate how the two types of swinging motions (the wobbling of the rod and the waggling of the spindle) couple. Choose different thread lengths, and determine that ratio of \(\displaystyle L/h\) at which the period of the alternation between the two types of oscillation is the longest.

(6 points)

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Problems with sign 'P'

Deadline expired on 10 December 2015.

P. 4768. How long does the Earthraise take on the Moon?

(3 points)

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P. 4769. The motion of points \(\displaystyle A\) and \(\displaystyle B\) is determined - but otherwise not restricted. How can the velocity and the acceleration of the midpoint of the line segment \(\displaystyle AB\) be expressed in terms of the appropriate data of points \(\displaystyle A\) and \(\displaystyle B\)?

(4 points)

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P. 4770. A child of mass \(\displaystyle m\) is bouncing on a trampoline. What is the average acceleration of the child during one complete jump (up and down motion)?

(3 points)

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P. 4771. A graduated cylinder of cross-section \(\displaystyle 50~\rm cm^2\) is filled with some salt solution of density 1.2 g/cm\(\displaystyle {}^3\) up to a height of 10 cm, and then carefully (without mixing the layers) 10 cm-high clean water is stratified onto it.

\(\displaystyle a)\) What is the pressure at the bottom of the cylinder?

\(\displaystyle b)\) What is the total gravitational potential energy of the two samples of liquid with respect to the bottom of the cylinder?

By what amount will these quantities change if the two samples of liquid are mixed? (Neglect the change in the volume of the whole mixture due to the mixing process.)

(4 points)

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P. 4772. A negligible mass and homogeneous spring spring of spring constant \(\displaystyle D\) and of length \(\displaystyle L\) is cut into \(\displaystyle n\) pieces. The small pieces of the spring are alternately attached to bodies of mass \(\displaystyle m\) each, thus creating a chain (at its end there is a body of mass \(\displaystyle m\)). Then the chain is hung.

\(\displaystyle a)\) How should the spring be cut, in order to gain the same lengths of springs in the chain when it is hanging at rest?

\(\displaystyle b)\) What distance does the centre of gravity move down if the bottom of the chain is pulled down by a distance of \(\displaystyle \delta\)?

(Let for example \(\displaystyle DL=mg\) and \(\displaystyle n=5\).)

(5 points)

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P. 4773. A uniform thin rod of length \(\displaystyle L=2\) m is suspended with a hook at the ceiling. The rod is raised into horizontal position and it is released with zero initial velocity. After passing the vertical position, when the rod encloses an angle of \(\displaystyle 30^\circ\) with the vertical, the rod leaves the hook.

\(\displaystyle a)\) What is the least distance between the ground and the hook, if the rod arrives to the ground in a vertical position?

\(\displaystyle b)\) What is the maximum height of the bottom end of the rod during the motion?

(5 points)

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P. 4774. A bumper is mounted to a vertical rod, hinged at its bottom end, at a distance of \(\displaystyle d\) from the hinge. There is a small bead at rest on the bumper. The rod is started to vibrate and executes SHM about its original position with a small angular amplitude of \(\displaystyle \theta_0\), as shown in the figure. What should the frequency of the vibration be in order that the bead flies off? (Friction is negligible.)

(5 points)

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P. 4775. Thin copper and iron strips are riveted together along their length to form a bimetallic strip. The distance between the two strips is 1 mm. The bimetallic strip is flat at the temperature of \(\displaystyle 0\;{}^\circ\)C. What will the radius of the bent strip be at a temperature of \(\displaystyle 200\;{}^\circ\)C?

(4 points)

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P. 4776. What is the resistance \(\displaystyle R_5\) of the bridge circuit shown in the figure if the equivalent resistance between points \(\displaystyle A\) and \(\displaystyle B\) is 9 \(\displaystyle \Omega\)? Data: \(\displaystyle R_1=5~\Omega\), \(\displaystyle R_2=12~\Omega\), \(\displaystyle R_3=15~\Omega\), \(\displaystyle R_4=8~\Omega\).

(4 points)

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P. 4777. At a radar station where electromagnetic impulses of 20 cm wavelength are emitted, a difference of 2778 Hz is detected between the frequency of the emitted and reflected impulses. What was the velocity of the air-plane which reflected the emitted beam?

(4 points)

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P. 4778. We would like to enhance the efficiency of a rectangular solar PV panel with using two plane mirrors. The mirrors fit along the whole length of the two longer edges of the panel at a sort of optimal angle. Consider the situation when the incident solar rays are nearly perpendicular to the panel. Our aim is to achieve that all the rays coming from the Sun and reflected from the mirrors should fall onto the panel thus the illumination of the panel is doubled.

What should the optimal angle and the width of the mirrors be if the material used is to be minimum?

(5 points)

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P. 4779. At a height of \(\displaystyle h\) above a broad grounded metal sheet there is a metal sphere of radius \(\displaystyle R\) in the air (\(\displaystyle R\ll h\)). The sphere is joined to the sheet with a very thin piece of metal wire. A point charge is brought near the sphere, such that its distance from both the sphere and the sheet is \(\displaystyle h\). Then we first remove the wire and after the charge. What will the potential difference between the sphere and the sheet be?

(6 points)

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Send your solutions to the following address:

    KöMaL Szerkesztőség (KöMaL feladatok),
    Budapest 112, Pf. 32. 1518, Hungary
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