BIO Leg Raises. In a simplified version of the musculature action in leg raises, the abdominal muscles pull on the femur (thigh bone) to raise the leg by pivoting it about one end (?Fig. P11.53?). When you are lying horizontally, these muscles make an angle of approximately 5° with the femur, and if you raise your legs, the muscles remain approximately horizontal, so the angle ? increases. Assume for simplicity that these muscles attach to the femur in only one place, 10 cm from the hip joint (although, in reality, the situation is more complicated). For a certain 80-kg person having a leg 90 cm long, the mass of the leg is 15 kg and its center of mass is 44 cm from his hip joint as measured along the leg. If the per-son raises his leg to 60° above the horizontal, the angle between the abdominal muscles and his femur would also be about 60°. (a) With his leg raised to 60°, find the tension in the abdominal muscle on each leg. Draw a free-body diagram. (b) When is the tension in this muscle greater: when the leg is raised to 60° or when the person just starts to raise it off the ground? Why? (Try this yourself.) (c) If the abdominal muscles attached to the femur were perfectly horizontal when a person was lying down, could the person raise his leg? Why or why not?
Solution 57P Introduction First we have to draw the free body diagram. Then we have to calculate the tension of the abdominal muscle. Then we have to discuss whether the tension increases or decreases if angle increases. Finally we have to discuss whether it is possible or not to lift leg if the abdominal muscle remains horizontal when the leg is horizontal. Step 1 The free body diagram of the situation is shown below. Now let us consider the moment of the forces about the hip joint. Since leg is stable there, the total moment will be zero. So we have r T sin r wcos = 0 1 2 r T1sin = r wc2s w cos 2 mg cos (0.44 m)(15 kg)(9.8 m/s ) cos(60°) T = sin = r sin = (0.10 m) sin(60°) = 373.4 N 1 The tension on the abdominal muscle from each leg is 373.4 N. Step 2 When the leg rises to the 60°, then the component of the tension perpendicular to the leg is greater and the component of weight perpendicular to the leg is small. Hence the required tension will be less but when the leg is horizontal the component of the weight perpendicular to the leg is very high and the component of the tension perpendicular to the leg will be very small. Hence the required force tension will be large. So the tension is greater when the leg rises from the ground.