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# o_q - In this problem we deal with a simplified model of...

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1 In this problem we deal with a simplified model of accelerometers designed to activate the safety air bags of automobiles during a collision. We would like to build an electromechanical system in such a way that when the acceleration exceeds a certain limit, one of the electrical parameters of the system such as the voltage at a certain point of the circuit will exceed a threshold and the air bag will be activated as a result. Note: Ignore gravity in this problem. 1 Consider a capacitor with parallel plates as in Figure 1. The area of each plate in the capacitor is A and the distance between the two plates is d . The distance between the two plates is much smaller than the dimensions of the plates. One of these plates is in contact with a wall through a spring with a spring constant k , and the other plate is fixed. When the distance between the plates is d the spring is neither compressed nor stretched, in other words no force is exerted on the spring in this state. Assume that the permittivity of the air between the plates is that of free vacuum 0 ε . The capacitance corresponding to this distance between the plates of the capacitor is d A C 0 0 ε = . We put charges Q + and Q - on the plates and let the system achieve mechanical equilibrium. Figure 1 1.1 Calculate the electrical force, E F , exerted by the plates on each other. 0.8 1.2 Let x be the displacement of the plate connected to the spring. Find x . 0.6 1.3 In this state, what is the electrical potential difference V between the plates of the capacitor in terms of Q , A , d , k ? 0.4 1.4 Let C be the capacitance of the capacitor, defined as the ratio of charge to potential difference. Find 0 C C as a function of Q , A , d and k .

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o_q - In this problem we deal with a simplified model of...

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