. A block having mass m and charge +Q is connected to a spring having constant k
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Question
. A block having mass m and charge +Q is connected to a spring having constant k. The block lies on a frictionless horizontal track, and the system is immersed in a uniform electric field of magnitude E, The block is released from rest when the spring is unstretched (at x = 0). (a) By what maximum amount does the spring expand? (Use Q, E, and k as necessary.) (b) What is the equilibrium position of the block? (Use Q, E, and k as necessary.) (c) Show that the block's motion is simple harmonic and determine its period. (Use Q, E, m, and k as necessary.) (d) Repeat part (a) if the coefficient of kinetic friction between block and surface isExplanation / Answer
Forces on block: spring force (F=-kX), Electric force (F=EQ*c, with c being the electric constant) Since F=ma, -kx+EQc=ma a=d/dt (d/dt x) I don't know how much you know about differential equations, but this one simplifies quite a bit, and x=Asin(sqrt(k/m)t+phi)+cEQ/k, with A and phi being arbitrary constants for boundary conditions. It's easy to solve for these by setting x(0)=0 and d/dt x(0)=0, although the question doesn't ask you too. So from this, you see that the block still moves with the sin(sqrt(k/m)t) type of motion typical for simple harmonic motion, and with the same period (sqrt m/k). The only difference is the equilibrium position, which will be shifted from x=0 be EQc/k units, and the amplitude (if you solved for A, you would get -EQc/k, and phi=pi/2).
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