Exercise:
Ein Gas Stoffmenge n sei bei einem Druck p in einem hohlen Zylinder mit A Querschnittsfläche durch einen sich reibungsfrei bewegen lassen Kolben eingeschlossen. Die in der Ruhelage vom Gas eingenommene Zylinderhöhe betrage h. Zeige durch Angabe von K in F-Kx dass der Kolben harmonisch schwingt falls man ihn leicht auslenkt. Hinweis: Universelle Gasgleichung pVnRT. center tikzpicture tikzstylepistonblue!!blacktop colorblue!!black!bottom colorblue!!black! middle colorblue!!black!shading angle tikzstylewalldarkblue!!blacktop colorblue!!black!bottom colorblue!!black!shading angl tikzstylewallblue!!blacktop colorblue!!black!bottom colorblue!!black!shading angle tikzset gasparticle/.pic tikzset/gasparticle/.cd# draw-Latexlengthwidthgreen!!blackthick -- vec; nodecirclefillinner sep.ball colorblack!!blue at ; /gasparticle/.search also/tikz /gasparticle/.cd vec/.store invec vec:. scope defRa. defRb. defra. defrb. defw. % wall thickness defx. % piston position defL. % container length defl. % piston arm length defv. % velocity % WALL drawwall Rb -- -Rb --++ L arc -::Ra and Rb -- cycle; drawwalldark ellipse Ra and Rb; % SHELL drawwalldark Rb rectangle++ Lw; drawwalldark -Rb rectangle++ L-w; drawwalldark LRb+w arc :-:Ra+w and Rb+w --++ w arc -::Ra and Rb -- cycle; drawwalldark Rb arc ::Ra and Rb --++ -w arc -:-:Ra+w and Rb+w -- cycle; drawwalldark xRb arc ::Ra and Rb --++ -*w arc -:-:Ra and Rb -- cycle; drawpiston x ellipse Ra and Rb; drawpiston xrb arc ::ra and rb --++ l --++ *rb -- cycle; drawwalldark x+l ellipse ra and rb; draw-thickblue!!black x.*Rb toin- x.*Rb nodeabove left A; drawthick colorgreen!!black|- -.--x-.; drawthick colorgreen!!black|- x-.---.;% noderight x; nodecolorgreen!!black at x-. h; % GAS PARTICLE pic at -.*x .*Rb gasparticlevec -:.*v; pic at -.*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec-:.*v; pic at .*x .*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec -:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec -:.*v; pic at .*x .*Rb gasparticlevec-:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec -:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec-:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec -:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec :.*v; scope scopexshiftcm defRa. defRb. defra. defrb. defw. % wall thickness defx. % piston position defxf. % piston position faded defL. % container length defl. % piston arm length defv. % velocity % WALL drawwall Rb -- -Rb --++ L arc -::Ra and Rb -- cycle; drawwalldark ellipse Ra and Rb; % SHELL drawwalldark Rb rectangle++ Lw; drawwalldark -Rb rectangle++ L-w; drawwalldark LRb+w arc :-:Ra+w and Rb+w --++ w arc -::Ra and Rb -- cycle; drawwalldark Rb arc ::Ra and Rb --++ -w arc -:-:Ra+w and Rb+w -- cycle; %PISTON drawwalldark xRb arc ::Ra and Rb --++ -*w arc -:-:Ra and Rb -- cycle; drawpiston x ellipse Ra and Rb; drawpiston xrb arc ::ra and rb --++ l --++ *rb -- cycle; drawwalldark x+l ellipse ra and rb; %faded PISTON drawwalldark opacity. xfRb arc ::Ra and Rb --++ -*w arc -:-:Ra and Rb -- cycle; drawpistonopacity. xf ellipse Ra and Rb; drawpiston opacity. xfrb arc ::ra and rb --++ l --++ *rb -- cycle; drawwalldark opacity. xf+l ellipse ra and rb; drawthick colorgreen!!black|- -.--x-.; drawthick colorgreen!!black|- x-.--.-.; drawthick colorgreen!!black|- .-.---.;% noderight x; nodecolorgreen!!black at x-. h-x; draw- colorred ..--x. nodeleft x; % GAS PARTICLE pic at -.*x .*Rb gasparticlevec -:.*v; pic at -.*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec-:.*v; pic at .*x .*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec -:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec -:.*v; pic at .*x .*Rb gasparticlevec-:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec -:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec-:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec -:.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x-.*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec :.*v; pic at .*x .*Rb gasparticlevec :.*v; scope tikzpicture center

Solution:
Es gilt: F -Delta p A -p-p_ A -leftfracnRTV-fracnRTV_right A -leftfracnRTAh-fracnRTAh_right A -nRTleftfrach-frach_right -nRTleftfrach_-x-frach_right -nRTfracxh_h_-x -fracnRTh_^x