$(8$ pts $)$ The knee flexor $($hamstring$)$ muscles insert on the tibia at an angle of $18$ at a distance of $6$ cm from the knee joint center $($see figure below$).$ The lower leg $($including the foot$)$ weighs $118$ N and has a moment arm of $15$ cm from the axis of rotation at the knee. In this position, the knee flexors exert a pulling muscle force of $453$ N $.$

a$.$ What is the moment arm of the knee flexor muscles?

b$.$ What is the total torque contributing to flexing the leg?

c$.$ If the knee extensors have an angle of insertion of $25$ and a moment arm of $2$ cm in this position, what is the minmum amount of knee extensor muscle force needed to begin extending the leg?

d$.$ In the static position, what is the total muscle force $($from flexors and extensors$)$ contributing to stabilizing the limb$?$ To rotating the limb$?$

e$.$ There is another knee flexor, the gastronemius, that originates above the knee at the distal end of the femur and inserts at an angle of $7$ on the calcaneus $($heel bone$)25$ cm from the axis of rotation at the knee. Draw and label the gastrocnemius muscle force on the figure.

f$.$ If the magnitude of total torque contributing to flexing the leg is now $15\%$ greater than it was in part B$,$ what is the muscle force of the gastrocnemius? Everything else remains the same.

a$.$ Moment arm of knee flexors:

b$.$ Torque contributing to flexing leg:

c$.$ Minimum force to extend leg:

d$.$ Total muscle force for rotating:

For stabilizing:

f$.$ Muscle force of gastrocnemius: