International Physics Olympiad · Year 2013

Problems

Problem 1In an insulated vessel, $m_h=0.30\,\mathrm{kg}$ of water at $70^\circ\mathrm C$ is mixed with $m_c=0.20\,\mathrm{kg}$ of water at $10^\circ\mathrm C$ . Water has specific heat $c=4200\,\mathrm{J\,kg^{-1}K^{-1}}$ ; neglect the vessel heat capacity and heat loss. After mixing, the water reaches one uniform final temperature, with no evaporation or phase change. Heat is exchanged only between the two portions of water; the insulated vessel takes up no energy. Define heat as positive when the portion considered gains energy and negative when it loses energy. (a) Find the equilibrium temperature $T_f$ . (b) Calculate the signed heat transferred by the hot water. (c) Find the heat received by the cold water and check energy conservation. (d) State the ordering of $T_c,T_f,T_h$ .Solutions: 1
Problem 2Two masses $m_1=4.0\,\mathrm{kg}$ and $m_2=6.0\,\mathrm{kg}$ hang from a light inextensible string over a massless frictionless pulley; the system starts from rest. Take $g=10\,\mathrm{m\,s^{-2}}$ . The string stays taut and does not slip on the pulley; its free rotation makes the tensions in the two segments equal. The masses move only vertically, so the lighter mass rises by the same distance and at the same speed that the heavier mass descends. Consider the motion after release. (a) Find the acceleration of each mass and the string tension. (b) Determine their speeds after $3.0\,\mathrm s$ . (c) How far has each mass moved by then?Solutions: 1
Problem 3A ball is launched horizontally from a cliff of height $h=20\,\mathrm m$ with initial speed $u=20\,\mathrm{m\,s^{-1}}$ . Neglect air resistance and take $g=10\,\mathrm{m\,s^{-2}}$ . Choose the launch point as the origin, with the horizontal axis forward and the vertical axis upward. The ground is a vertical distance $h$ below the launch point; treat the ball as a point particle that hits no other object before landing. Measure time from the instant it leaves the cliff. (a) Find the time to reach the ground and the horizontal range. (b) Determine the velocity components just before impact. (c) Find the impact speed and its angle below the horizontal.Solutions: 1