Olympiad problems

2009 F = Ma, 21

Tags: 2009 , problem 21

What is the value of the gravitational potential energy of the two star system? (A) $-\frac{GM^2}{d}$ (B) $\frac{3GM^2}{d}$ (C) $-\frac{GM^2}{d^2}$ (D) $-\frac{3GM^2}{d}$ (E) $-\frac{3GM^2}{d^2}$

The gravitational self potential energy of a solid ball of mass density $\rho$ and radius $R$ is $E$. What is the gravitational self potential energy of a ball of mass density $\rho$ and radius $2R$? (A) $2E$ (B) $4E$ (C) $8E$ (D) $16E$ (E) $32E$

2008 F = Ma, 21

Tags: 2008 , problem 21

Consider a particle at rest which may decay into two (daughter) particles or into three (daughter) particles. Which of the following is true in the two-body case but false in the three-body case? (There are no external forces.) (a) The velocity vectors of the daughter particles must lie in a single plane. (b) Given the total kinetic energy of the system and the mass of each daughter particle, it is possible to determine the speed of each daughter particle. (c) Given the speed(s) of all but one daughter particle, it is possible to determine the speed of the remaining particle. (d) The total momentum of the daughter particles is zero. (e) None of the above.

2011 F = Ma, 21

Tags: 2011 , problem 21

An engineer is given a fixed volume $V_m$ of metal with which to construct a spherical pressure vessel. Interestingly, assuming the vessel has thin walls and is always pressurized to near its bursting point, the amount of gas the vessel can contain, $n$ (measured in moles), does not depend on the radius $r$ of the vessel; instead it depends only on $V_m$ (measured in $\text{m}^3$), the temperature $T$ (measured in $\text{K}$), the ideal gas constant $R$ (measured in $\text{J/(K} \cdot \text{mol})$), and the tensile strength of the metal $\sigma$ (measured in $\text{N/m}^2$). Which of the following gives $n$ in terms of these parameters? (A) $n=\frac{2}{3}\frac{V_m\sigma}{RT}$ (B) $n=\frac{2}{3}\frac{\sqrt[3]{V_m\sigma}}{RT}$ (C) $n=\frac{2}{3}\frac{\sqrt[3]{V_m\sigma^2}}{RT}$ (D) $n=\frac{2}{3}\frac{\sqrt[3]{V_m^2\sigma}}{RT}$ (E) $n=\frac{2}{3}\sqrt[3]{\frac{V_m\sigma^2}{RT}}$

2009 F = Ma, 21

2010 F = Ma, 21

2008 F = Ma, 21

2011 F = Ma, 21