Answer Problem 1 correctly and include/show ALL work used to find solutions.

2.3 SPINE FIN OF A CATFISH Although in most cases good lubrication of bonecontact surfaces is essential1 there are a few cases in nature where hone contacts are purposer unluhricated to increase friction. The catfish has such a joint connecting its dorsal spine n to the rest ofits skeleton (Fig. 13}. Normally the n is folded at against the tho-d};1 but when the fish is attacked. the appropriate muscles pull The bone of the fin into a space provided in the underlying skeleton. Since the coefficient of friction between the n bone and the skeleton is high, the frictional force tends to lock the fin in the up position. In order to remove The n, a force must be applied in a predominantly vertical direction with respect to the underlying skeleton. The erect sharp fin discourages predators from eating the catfish. Figure 2.3h is a simplified representation of the spine and The protruding fin. The shaded block represents The movable fin bone... and the horizontal block is the spine holding the fin. Assume that a force F at an angle 19 is applied at point A to dislodge the bone. The force is shown to act at pointA, 2.5 cm above point B. Dorsal spine {A} Fin bone III'1 {a} ' 1 \"m ' FIGURE 2.3 {A} lCatfish. {B} Simplified representation of the spine in line catsh. Problem 1: Referring to Figure 2.3 in the textbook, assume that the angle between the fin bone and the spine is 45 and the coefficient of friction between the bones to prevent dislodging of the bone is 1.0, what is the value of the angle 0 at which a force of 0.2 N will just dislodge the bone? What would this angle be if the bones were lubricated (u = 0:01)?Dorsal spine (A) 2.5 cm Fin bone 1/2 cm Spine B C (B) 1 cm FIGURE 2.3 (A) Catfish. (B) Simplified representation of the spine in the catfish. 26 Physics in Biology and Medicine The dimensions shown in the figure are to be used in the calculations required for Exercise 2-3. The applied force tips the bone, and as a result reaction forces are set up at points B and C. The components of these forces normal to the fin-bone surface produce frictional forces that resist removal of the bone. Calculation of some of the properties of the locking mechanism is left as an exercise