A 4.00-kg block carrying a charge Q = 50.0 mu C is connected to a spring for which k = 100 N/m. The block lies on a frictionless horizontal track, and the system is immersed in a uniform electric field of magnitude E = 5.00x10^5 V/m, directed as shown in Figure. If the block is released from rest when the spring is unstretched (at x = 0), (a) by what maximum amount does the spring expand? (b) What is the equilibrium position of the block? (c) Show that the block’s motion is simple harmonic, and determine its period. (d) Repeat part (a) if the coefficient of kinetic friction between block and surface is 0.200.

Today we have one of those beautiful problems where we suffer because the teacher asking us was remember everything you learned about oscillations in physics I to solve a problem of physics II such problems are very nice (at least so say the teachers xD) to solve something like this simply we must be organized understand what the problem asks and how to obtain it It's fundamental.
Enough of such talk and let's start with what we are concerned what will we do in the part a? think... that's the first! thinking to understand that the maximum elongation of the spring is the amplitude we can obtain the amplitude of the fact that this occurs in the expression of maximum elastic potential energy and that this energy level is reached when the electric potential energy with the elastic potential energy are equal see how serious this:

Part b. It is simpler what means equilibrium ? equilibrium is reached when the forces acting on the body are balanced in words, they are equal (the truth is more complicated than this. the definition that we gave it's to the mechanical balance and lack to say that the sum of all torques is zero. There are different types of balance as appropriate, mechanical, thermodynamic, electrical , etc. and which have the sub branches of stable equilibrium and unstable equilibrium at this level we will comply with understand that equilibrium is reached when the sum of the forces and torques is zero) see the solution:

Part c. Here children of men differ. children go to his mom to mourn. we are mens and say to demonstrate that it is a simple harmonic motion we must make the free-body diagram of the block, obtain the equation of motion that governs it and show that the equation of motion is a simple harmonic oscillator equation. see if we can do it

we was demonstrated the fulfillment of simple harmonic motion. only lack find the period. we have defined a basic amount and is the natural frequency of system. with this we can get the period from the following expression:

Part d. here we did the job previously we did in the part a, that changes? Now when there was displacement block to maximum elongation part of the energy is lost as heat. Heat defined as the work done by the friction force. Ee then just add this little detail in energy balance on part a to obtain the desired value:


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