$2.$ Consider a muscle$/$tendon complex as shown in Figure $2.$ To find out how much the tendon and the muscle are extended when the complex as a whole is loaded with a force $\backslash ($ F $\backslash ),$ a very crude two$-$bar model can be used. The muscle is modeled as a bar with length $\backslash ($ l$\_\{1\}\backslash ),$ Young's modulus $\backslash ($ E$\_\{1\}\backslash )$ and cross section $\backslash ($ A$\_\{1\}\backslash ).$ At point $\backslash ($ B $\backslash )$ the muscle is attached to the tendon, which is modeled as a second bar with length $\backslash ($ l$\_\{2\}\backslash ),$ Young's modulus $\backslash ($ E$\_\{2\}\backslash )$ and cross section $\backslash ($ A$\_\{2\}\backslash ).$ At point $\backslash ($ A $\backslash )$ the muscle is attached to the bone, which we consider as a rigid fixation. At point $\backslash ($ C $\backslash )$ a force $\backslash ($ F $\backslash )$ is applied in the direction of the bar $($tendon$).$

Fig. $2.$ Question $2$

$($a$)$ Determine the internal force in a cross section between $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ and in a cross section between $\backslash ($ B $\backslash )$ and $\backslash ($ C $\backslash ).$

$($b$)$ Determine the stress $\backslash (\backslash$sigma $\backslash )$ in a cross section between $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash )$ and a cross section between $\backslash ($ B $\backslash )$ and $\backslash ($ C $\backslash ).$

$($c$)$ What happens with the calculated forces and stresses if the Young's moduli of both muscle and tendon are reduced to half of their original value?

$($d$)$ Determine the displacements at point $\backslash ($ B $\backslash )$ and $\backslash ($ C $\backslash )$ as a result of the applied load $\backslash ($ F $\backslash )$ at point $\backslash ($ C $\backslash ).$