Съдържание

IPhO 2007, Теория — Задача 1.2

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IPhO 2007

ORANGE — 38th International Physics Olympiad, Isfahan-Iran, July 2007

Условие

Figure 2, shows a mass $M$ which is attached to a conducting plate with negligible mass and also to two springs having identical spring constants $k$ . The conducting plate can move back and forth in the space between two fixed conducting plates. All these plates are similar and have the same area $A$ . Thus these three plates constitute two capacitors. As shown in Figure 2, the fixed plates are connected to the given potentials $V$ and $-V$ , and the middle plate is connected through a two-state switch to the ground. The wire connected to the movable plate does not disturb the motion of the plate and the three plates will always remain parallel. When the whole complex is not being accelerated, the distance from each fixed plate to the movable plate is $d$ which is much smaller than the dimensions of the plates. The thickness of the movable plate can be ignored.

A box accelerating to the right (arrow a) contains a mass M (dotted block) held between two springs k attached to the box walls; three vertical plates hang inside: two fixed plates (labelled Fixed Plate on both sides) and the movable middle plate attached to M (labelled Movable plate). Below the box, the left fixed plate goes to a battery V and the right fixed plate to a battery V, both returning to a grounded common wire; the movable plate goes to a two-state switch with contacts α (to ground) and β (to ground through a capacitor C_S). — Figure 2

The switch can be in either one of the two states $\alpha$ and $\beta$ . Assume that the capacitor complex is being accelerated along with the automobile, and the acceleration is constant. Assume that during this constant acceleration the spring does not oscillate and all components of this complex capacitor are in their equilibrium positions, i.e. they do not move with respect to each other, and hence with respect to the automobile.
Due to the acceleration, the movable plate will be displaced a certain amount $x$ from the middle of the two fixed plates.

2 Consider the case where the switch is in state $\alpha$ i.e. the movable plate is connected to the ground through a wire, then

2.1 Find the charge on each capacitor as a function of $x$ . [0,4 т.]

2.2 Find the net electrical force on the movable plate, $F_E$ , as a function of $x$ . [0,4 т.]

2.3 Assume $d \gg x$ and terms of order $x^2$ can be ignored compared to terms of order $d^2$ . Simplify the answer to the previous part. [0,2 т.]

2.4 Write the total force on the movable plate (the sum of the electrical and the spring forces) as $-k_{eff}\,x$ and give the form of $k_{eff}$ . [0,7 т.]

2.5 Express the constant acceleration $a$ as a function of $x$ . [0,4 т.]

Отговори

Решение

Оригинал в Архива: 1.pdf · официални решения: 1_sol.pdf