[別館]球面倶楽部零八式markIISR

東大入試数学中心。解説なので解答としては不十分。出題年度で並ぶようにしている。大人の解法やうまい解法は極めて主観的に決めている。

1921-01-01

1921年(大正10年)東京帝國大學理學部(物理科)數學及び力学(全5問うち力学2問)

[1] Calculate the value of $\log_e 1.5$ to $3$ decimal places.

[2] Shew that the function $u=A\dfrac{\sin kr}{r}$ satisfies the differential equation
$\dfrac{\delta^2 u}{\delta x^2}+\dfrac{\delta^2 u}{\delta y^2}+\dfrac{\delta^2 u}{\delta z^2}+k^2 u=0$ ,
where $k$ and $A$ are constants, and $r=\sqrt{x^2+y^2+z^2}$ .

[3] Find the direction cosines of the straight line defined by $a_1x+b_1y+c_1z=d_1, \quad a_2x+b_2y+c_2z=d_2$ .

[4]（力學） A shot of mass $m$ is fired from a gun of mass $M$ with velocity $u$ relative to the gun; find the velocities of the shot and the gun relative to the ground.

[5]（力學）Three forces $F_1$ , $F_2$ , $F_3$ act on a solid triangular lamina along its sides as shown in the figure. If there are relations $\dfrac{F_1}{\overline{AB}}=\dfrac{F_2}{\overline{BC}}=\dfrac{F_3}{\overline{CA}}$
is the system of the forces in equilibrium?