112年 - 112 國立臺灣大學_碩士班招生考試試題：普通物理學.pdf#138487

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二、計算題
1. A homogeneous thin half circle of radius R with a total mass M is symmetric about the y axis, as shown in Figure 5(a). (a) Find the coordinate of the center of mass (COM), (x_COM, y_COM). (b) Find the rotation inertia when it rotates around the axis passing through its COM (in parallel with the z-axis). (c) If the half circle is rotated in the xy plane with a small angle θ on a frictionless pivot at point P which is on the y axis, as shown in Fig. 5(b), and released. Show that the half circle will execute simple harmonic motion, and find the period T of simple harmonic motion, where the direction of gravitational acceleration g is in −ŷ. (15%)

2. A uniform insulating rope of mass M and length L hangs from a ceiling. A small metal ball with charge +Q and mass m is suspended at the bottom of the rope, which is above an infinite grounded plane at a distance d from it, as shown in Figure 6. There are surface charges induced in the grounded plane. (a) Find the electric force acting on the charged metal ball. (b) The surface charge density at point O, which is directly below the ball. (c) Find the speed of a transverse wave on the rope due to the electric and gravitational forces, which is a function of y, the distance from the lower end. (15%) [Hint: the speed of a transverse wave on a string v = with tension τ and linear mass density μ.]

3. Try to derive the wave equation of the electromagnetic wave from Maxwell's equations, and obtain that its propagation speed is the speed of light c in vacuum. (10%)