BME 207 Introduction to Biomechanics Spring Homework 1

Transcription

1 January 19, 2018 UNIVERSITY OF RHODE ISLAND Deartment of Electrical, Comuter and Biomedical Engineering BME 207 Introduction to Biomechanics Sring 2018 Problems in the textbook. Homework 1 Problem 18 The four colanar vectors shown in the figure have the following magnitudes (in cm): A = 4 B = 7 C = 3 D = 5 A. Comute the x and y comonents of each vector. B. Write each vector in terms of the coordinate unit vectors î and ĵ (in the x and y directions, resectively). C. Comute the vector E = A + D. D. Comute the vector F = A + B + C + D. E. Comute the magnitude of F. F. Comute the angle between F and the x axis. G. Comute the smallest angle between E and F. y C B A x D Answers B. A = 4î cm, B = 6.06î ĵ cm, C = 3ĵ cm, D = 3.54î 3.54ĵ cm C. E = 0.464î 3.54ĵ cm D. F = 6.52î ĵ cm E. F = 7.17 cm F G

2 Problem 19 The two colanar vectors have the magnitudes A = 7 m and B = 5 m. y 15 B A 25 x Comute the: A. unit vectors in the direction of A and B; B. dot roduct s = A B; ( C. dot roduct t = A ) ( B ) ; D. cross roduct C = A B; E. cross roduct D = B A. Answers A. û A = 0.906î ĵ m, û B = 0.259î ĵ m B. s = m 2 C. t = m 2 D. C = ˆk m 2 E. D = ˆk m 2-2 -

3 Problem 20 The figure shows two forces, E and F, alied to a bracket on a hysical theray machine. Determine the angle θ at which E must be alied to have the combined effect of E and F be equal to that of a single 200 N force. 70 E = 100 N θ F = 150 N Answers or Problem 21 A force vector F = 7î + 9ĵ + 3ˆk N has a line of action that asses through oint, which has coordinates of x = 3, y = 6, and z = 5 (all in cm). A. A second oint q is located at coordinates x = 1, y = 2, and z = 2 (cm). Comute the comonents of the unit vector ˆr that originates at q and is directed at. B. Comute the cross roduct M = ˆr F. C. Comute the angle between ˆr and each of the three coordinate axes. D. Find F, the comonent of F that is arallel to ˆr. E. Find F, the comonent of F that is orthogonal (erendicular) to ˆr. Check your answer by showing the dot roduct of F with ˆr is zero, and F + F = F. F. Comute the vector U that is orthogonal (erendicular) to the lane containing vector ˆr and vector F. Answers A. ˆr X = 0.371, ˆr Y = 0.743, ˆr Z = cm B. M = 2.785î 5.014ĵ ˆk N-cm C. Angle between ˆr and the x axis: y axis: z axis: D. F = 2.138î ĵ ˆk cm E. F = 9.138î ĵ 0.207ˆk cm F. U = 2.785î 5.014ĵ ˆk N-cm - 3 -

4 Problem 22 Zoologists estimate the jaw of a lion is subjected to a force P as large as 800 N. What forces T and M must be exerted by the temoralis and masseter muscles, resectively, to suort this value of P? Answers M = N, T = N - 4 -

5 Problem 23 The left figure shows an athlete executing a warm-u exercise while holding a 5 lb weight in each hand. The athlete s shoulders are arallel with the x axis, and the lower right leg and torso are arallel with the y axis. The idealized stance is shown on the right; tables below give the lengths and angles of the body segments. A. Comute the x,y coordinates for the location of each weight, W R and W L. B. Comute the osition vector P from the origin (x = 0, y = 0) to the center of the shoulders. C. Comute the cross roduct of the osition vector P with the force vector W L. D. Sketch the five external forces, with their roer directions, acting on the athlete s body. Assume the athlete s weight is concentrated at the midoint of the torso. Just sketch the forces do not comute their magnitudes. φ D E F γ W L θ A W R y y β B α C x x Lengths (inches) torso: A = 24 neck-shoulder: D = 8 thigh: B = 18 uer arm: E = 10 lower leg: C = 21 lower arm: F = 13 Angles (degrees) α = 56 θ = 43 β = 51 φ = 18 γ = 37 Answers A. W R is located at x = 5.3, y = 43.7 inches; W L at x = 40.4, y = 70.2 inches B. P = 14î ĵ inches C. P WL = 70ˆk lbf-in - 5 -

6 Problem 24 To oerate roerly the traction bar shown in figure (i) requires a torque T with a certain magnitude about oint. The torque is created by the 20 N force F alied at oint a as shown in figure (ii). F = 20 N (i) c (ii) 4 cm a b 8 cm a 4 cm 7 cm (iii) G (iv) H c b I J (v) 45 c (vi) c 30 A. Comute the magnitude and direction of the torque T about due to the force F in figure (ii). B. If force F is relaced by force G, which is located at oint b as shown in figure (iii), comute the magnitude of G required to roduce the same torque T. Why is G different from F? C. If the force F is instead relaced by force H located at oint c as shown in figure (iv), comute the magnitude of H required to roduce the same torque. How does the magnitude of H comare with that of F and G? D. Figure (v): now F is to be relaced by force I, which is oriented 45 u from the horizontal. Comute the magnitude of I required to roduce the same torque. How does the magnitude of I comare with that of F and G? Why are they different? E. Figure (vi): comute the torque Q about oint due to the force J = 56 N. Force J is much larger than the forces above; why isn t the magnitude of the torque Q also much larger? Answers A. T = 0.80 N-m (ccw); B. G = 11.4 N; C. H = 20 N; D. I = N; E. Q = 0.82 N-m (cw) - 6 -

7 Problem 25 A man has a body mass of m = 70 kg and an average density of 1.1 g/cm 3. A. Comute the man s volume. B. If the man is modeled as a shere with the same volume, comute his radius and diameter. C. If the man is modeled as a right circular cylinder with a height h = 172 cm, find the radius and diameter of the cylinder. D. If the man (with the same height) is modeled as a rectangular solid with a square cross-section, find the length of a side of the square. E. Reeat art D if the square cross-section is instead a rectangular cross-section, where the long and short cross-section dimensions have a ratio of either 3:1, 5:1, or 7:1. F. An allometric formula to estimate a erson s body surface area A (in cm 2 ) from their body mass m (kg) and height h (cm) is a : A = m h For each case above (arts B-E), comute the surface area of the geometric model and comare it to the man s body surface area as redicted by the formula. Which geometric model is best in terms of aroximating the body surface area? G. What is the ercent error in body surface area of the best geometric model versus the allometric formula? Answers A. volume = 63,636 cm 3 B. radius = 24.8 cm C. radius = 10.9 cm D. length = 19.2 cm E. For the 3:1 ratio the short-side length is cm F. The emirical formula gives body surface area = 18,251 cm 2. For the 3:1 ratio model the surface area is 16,020 cm 2. The 5:1 ratio model is best in terms of surface area. G. The 5:1 ratio model has 1.31% error. a D. DuBois and E.F. DuBois: A formula to estimate the aroximate surface area if height and weight be known. Archives of Internal Medicine 17: (June 1, 1916)

8 Problem 26 Consider a thin circular disk with a radius of r o = 1.5 inches. A. Comute, using roer unit conversions, the: i. area of the disk ii. erimeter of the disk iii. first moment of inertia given by BME Homework 1 January 19, 2018 Q = 2 3 r3 o iv. area moment of inertia using I = π 4 r4 o v. olar moment of inertia from J = π 2 r4 o B. An annulus is created from the disk above by drilling a 1.2 inch diameter hole in the center. Comute the: i. area of the annulus ii. erimeter of the inner edge iii. first moment of inertia given by Q = 2 3 ( r 3 o r 3 i ) iv. area moment of inertia using v. olar moment of inertia from I = π ( ) r 4 4 o ri 4 J = π ( ) r 4 2 o ri 4 Answers A. area = 4,560 mm 2 erimeter = 239 mm first moment of inertia = 36.9 cm 3 area moment of inertia = 165 cm 4 olar moment of inertia = 3,309,937 mm 4-8 -