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February 2, 2016 UNIVERSITY OF RHODE ISLAND Department of Electrical, Computer and Biomedical Engineering BME 207 Introduction to Biomechanics Spring 2016 Homework 2 Problems 3.1 through 3.10 in the textbook. Problem 11 Many women complain of lower back pain during pregnancy. The figure highlights the forces acting on the third lumbar vertebra L3 in the lower spine. The extensor muscles run along the back of the spine and generate a force Fm to balance the weight of the chest and gut regions; this force acts 2 inches posterior to the center of the vertebra. A compressive force FC acts on the center of the vertebra, which also supports 55% of the total body weight. This fraction of the body weight WB acts 2 inches anterior to the center of the vertebra. During pregnancy the abdomen increases in weight by an additional 20 pounds; WP , the force representing this weight increase, acts 10 inches anterior to the center of the vertebra. Calculate the compressive and extensor muscle forces, FC and Fm respectively, before and during the pregnancy of a woman originally weighing 130 pounds. L3 0.55WB FC WP Answers before pregnancy: FC = 143 pounds Fm = 71.5 pounds during pregnancy: FC = 263 pounds Fm = 171.5 pounds -1- Fm BME 207 - Homework 2 February 2, 2016 Problem 12 A patient holds a 15 kg weight W in the right palm; the upper arm is vertical and the forearm is horizontal. The bicep is attached to the forearm at x = 4 cm from the long axis of the humerus. The weight acts 6 cm left of the fingertips. If the bicep force B is vertical, compute its magnitude and the magnitude of the reaction force R at the elbow joint. Use the anthropometric data for a 50th percentile female. Note: the figure on the right is not a complete free body diagram! R B 6 cm 4 cm W Answers B = 1456.9 N, R = 1295.6 N Problem 13 Now consider a more realistic scenario for the same patient above: instead of the bicep force acting vertically, it is oriented at an angle α = 13 ◦ from the vertical. Compute the magnitude of B and the magnitude and direction of R at the elbow joint, using the anthropometric data for a 50th percentile female. B α = 13 ◦ R 6 cm θ 4 cm W Answers B = 1495.2 N, R = 1338.5 N, θ = 75.5 ◦ -2- BME 207 - Homework 2 February 2, 2016 Problem 14 Using Table 2 and Figures 1 and 2 in the “Human Anthropometric Data,” compute the: A. vertical distance from the elbow to the chin of a 95th percentile male; B. vertical distance from the fingertips to the buttocks of a 5th percentile male; C. horizontal distance between the fingertips of a 50th percentile female when her arms are outstretched (as in the left picture below); D. horizontal distance between the fingertips and the vertical distance from the ground to the fingertips of a 50th percentile female when her arms are raised thirty degrees above the horizon (right picture, above). Answers A. 17.7 inches; B. 8.0 inches; C. 73.6 inches; D. 65.8 inches and 67.0 inches Problem 15 Verify the total body masses of the 50th percentile male and female in Table 3. Problem 16 A person holds a 20 kg package. The mass of the person’s upper body above the lumbosacral disc (between the L5-S1 vertebrae) is 45 kg. A. When standing erect, the upper body’s center of mass is horizontally 2 cm anterior to the disc, and the center of mass for the package is 30 cm anterior to the disc. Sketch the loads imposed on the lumbosacral disc by the upper body and the package. Compute the moment about the disc caused by the combined upper body and package load. What is this moment in lb-in? (In low-speed automobile collisions, lumbar disc failure occurs at a mean bending moment of 1239 lb-in.) B. When bending over, the upper body’s center of mass is 25 cm in front of the disc, and the center of mass for the package is 40 cm in front of the disc. Sketch the loads imposed on the disc by the upper body and the package in the bent position, and compute the bending moment about the disc caused by the combined upper body and package load. C. How does bending over affect the load on the lumbosacral disc? Is there any engineering basis for the expression “Don’t lift with your back”? Answers A. upright: M = 67.7 N-m B. bent: M = 188.8 N-m -3- BME 207 - Homework 2 February 2, 2016 Problem 17 The left figure shows the athlete holding 5 lb weights in each hand. The shoulders are parallel with the x axis, and the lower right leg and torso are parallel with the y axis. The idealized stance is shown on the right; the tables below give the lengths and angles of the body segments. Assume the body weight WB = 150 lb acts at the midpoint of the torso. A. Draw the free body diagram of all the forces acting on the athlete’s body. B. Compute the reaction forces FL and FR acting at the left and right feet. C. Compute the sum of the moments about . . . i. the left hand, at WL ii. the base of the neck iii. the right heel, at FR F E φ WL D γ A θ WR y y β B C α x Lengths (inches) torso: A = 24 neck-shoulder: D = 8 thigh: B = 18 upper arm: E = 10 lower leg: C = 21 lower arm: F = 13 Answers B. FL = 65 lb, FR = 95 lb -4- x Angles (degrees) α = 56 θ = 43 β = 51 φ = 18 γ = 37 BME 207 - Homework 2 February 2, 2016 Problem 18 A method for estimating the anatomical center of gravity is shown. The patient lays on a balance board which is supported by a scale and a fulcrum. The weight of the patient W is known, as is the weight B of the board. The combined weight of the patient and board is supported by a scale, which reads S, and a force at the fulcrum. The distances from the fulcrum to the scale xS and to the board’s center of gravity xB are known. xS xC xB S W B A. Sketch the free body diagram of the patient and board. B. Derive an expression for xC , the distance of the patient’s center of gravity from the fulcrum, in terms of B, S, W , xB , and xS . C. Estimate the distance from the fulcrum to the patient’s center of gravity if the: • patient is 5 feet 8 inches tall and weighs 140 pounds; • board is 6 feet long and weighs 7 pounds; • distance from the fulcrum to the board’s center of gravity is 8 inches; • distance from the fulcrum to the scale is 39 inches; • scale reads 30 pounds 14 ounces. D. Assuming the patient is a 50th percentile male and the fulcrum is located at the buttocks, how high off the ground is the patient’s center of gravity when standing? Answers B. In terms of the known quantities: xC = SxS − BxB inches W C. 8.2 inches D. 41.2 inches -5-