Practice Problems Kinetics – Part 1 1. Use the data in Tables 2.5 and 2.6 to answer the following about the average woman (BW=67 kg; Ht=169 cm): a. What is the weight of her upper arm? (17 N) b. What is the moment of inertia for her upper arm about a somersault axis passing through the shoulder joint? (0.0507 kgm2) 2. As part of a warm-up routine, an aerobics instructor (mass = 53 kg; height = 161 cm) performed arm circles. For this movement, the arm was kept extended and was rotated in a sagittal plane about the shoulder joint. The length of her upper arm was 23 cm; her forearm was 25 cm; and her hand (to the tip of the longest finger) was 19 cm. a. Use tables 2.1 and 2.3 to determine the moment of inertia of the arm about the shoulder joint. (0.240 kgm2) b. If she completed an arm circle in 325 ms, what was the average angular velocity of the movement? (19.3 rad/s) c. What was the linear velocity of her wrist? 3. The masses and moments of inertia of a set of golf clubs are presented in the table below. A close examination of this table reveals that, while the masses of the clubs increased from the No. 1 wood to the No. 10 iron, with only two exceptions, the corresponding moments of inertia decreased. Can you account for this seemingly strange result? Masses and Moments of Inertia of Golf Clubs About Transverse Axes 10 cm from the Grip End Club Mass Moment (kg) of Inertia (kg.m2) 1 wood 0.371 0.224 2 wood 0.381 0.228 3 wood 0.385 0.224 3 iron 0.419 0.212 4 iron 0.426 0.208 5 iron 0.436 0.210 6 iron 0.440 0.206 7 iron 0.445 0.205 8 iron 0.454 0.205 9 iron 0.458 0.204 10 iron 0.464 0.203 * Wilson Sporting Goods Company, Sam Snead (1979 model) 4. For each of the following four positions, draw in the moment arms for the two weight vectors relative to the knee joint. What does this tell you about resistive torque in weight training? What about resistive force? When you combine these facts with what you already know about muscle torque, what practical relevance does this information have? 5. The gymnast in the figure below is performing a straddle seat circle on the high bar as part of an uneven parallel bars exercise. a. If her weight is 490 N, the distance between her line of gravity and the center of the bar is 0.47 m, and her moment of inertia relative to an axis through the center of the bar is 15.5 kg.m2, what angular acceleration is she experiencing at the instant depicted? (Ignore the likely existence of a frictional torque exerted by the bar on the hands.) b. If you did not ignore the frictional torque exerted by the bar on the hands, would her angular acceleration be greater than or less than the value you calculated in ‘a’? Explain your answer. c. As a practical matter, since she clearly would like as large an angular acceleration as she can get to complete the circle with ease, can you see any way in which she might increase her angular acceleration at the instant shown? 6. A patient with a total-body mass of 78 kg performs a knee extension exercise while wearing a deLorme boot (4.5 kg). The patient has a standing height of 1.83 m. Suppose that the (x,y) coordinates (cm) for various landmarks are as follows: Landmark Coordinates Knee joint 0, 0 Ankle joint -1, -44 Shank CG 0, -19 Foot CG 8, -49 deLorme boot CG 6, -52 a. Use the regression equations of Zatsiorsky and Seluyanov (Table below) to estimate the mass of the shank and the foot. b. Use the following equation to estimate the moment of inertia of the system (shank + foot + deLorme boot) about the somersault axis of the knee. n I mi ri 2 i 1 Regression Equations Estimating Body Segment Masses and Locations of Centers of Gravity Mass (kg) coefficients CG location (% segment length) coefficients Segment B0 B1 B2 B0 B1 B2 Foot -0.8290 0.0077 0.0073 3.7670 0.0650 0.0330 Shank -1.5920 0.0362 0.0121 -6.0500 -0.0390 0.1420 Thigh -2.6490 0.1463 0.0137 -2.4200 0.0380 0.1350 Hand -0.1165 0.0036 0.0017 4.1100 0.0260 0.0330 Forearm 0.3185 0.0144 -0.0011 0.1920 -0.0280 0.0930 Upper arm 0.2500 0.0301 -0.0027 1.6700 0.0300 0.0540 Head 1.2960 0.0170 0.0143 8.3570 -0.0025 0.0230 Upper torso 8.2144 0.1862 0.0584 3.3200 0.0076 0.0470 Middle torso 7.1810 0.2234 -0.0663 1.3980 0.0058 0.0450 Lower torso -7.4980 0.0976 0.0490 1.1820 0.0018 0.0434 Note. The multiple regression equations are in the form Y = B0 + B1 X1 + B2 X2 , where Y = predicted segment mass or CG location, = total body mass (kg), = height (cm), and = B 0 , B1, B2 = coefficients given in the table. Body segment mass and CG locations are expressed as functions of body mass and height. Adapted from Zatsiorsky and Seluyanov, 1983, & Enoka, 1994. 8. A 90 kg ice hockey player collides head-on with an 80 kg ice hockey player. If the first player exerts a force of 450 N on the second player, how much force is exerted by the second player on the first? 9. Draw two free body diagrams -- one of a person performing a push-up with the elbows fully extended and one of a person performing a push-up with the body in its lowest position. Is one of these positions easier to maintain? Why or why not? 10. A weight lifter performing a curl with a 300 N barbell stops the lifting motion with his forearms in a horizontal position. (A curl is an arm exercise performed in the standing position and in which the barbell is raised from a position with the elbows extended and the barbell resting against the performer's thighs to one with the arms flexed and the barbell under his chin.) a. If the line of action of the weight of the barbell is 30 cm from the axis of rotation, is the magnitude of the moment that the elbow flexors must exert to maintain position? (Ignore the weight of the weight lifter's forearms and hands in making your calculation.) b. If the weight of the weight lifter's forearms and hands were taken into account, would the moment that the elbow flexors had to exert be more or less than that calculated in part (a)? c. If the only elbow flexor muscle active were the biceps—a rather unlikely event—and the distance from its insertion (the radial tuberosity) to the axis of rotation was 3 cm, what vertical force would the muscle have to exert to maintain the forearm in its horizontal position? (Once again, ignore the weight of the liher's forearms and hands.) d. Under the conditions described, would the total force exerted by the biceps be more than, equal to, or less than that calculated in part (c)? e. How would the resistive force and torque change if the forearm were lowered to 45 degrees below the horizontal? 10. A photograph was taken of an athlete performing rehabilitation from knee surgery. The exercise was performed with the knee joint stationary so that the center of gravity of the system did not experience a linear acceleration. The shank-foot (Fw,s) weighed 120 N, and an ankle weight (Fw,a) of 250 N was attached. The distance between the knee joint and Fw,s was 15 cm, and the distance between the knee joint and Fw,a was 40 cm. a. What was the magnitude of the load torque in this position? b. If the leg was being lowered, i. What muscle group would be controlling the movement? ii. What is the maximum torque (Tm) that the muscle group may exert and still allow the leg to be lowered? c. Suppose the moment of inertia of the system about the knee joint (K) has a magnitude of 0.11 kg.m. What magnitude of torque (Tm) must the athlete exert to obtain a system acceleration of 9.4 rad/s? 11. Two muscles develop tension simultaneously on opposite sides of a joint. Muscle A, attaching 3 cm from the axis of rotation at the joint, exerts 250 N of force. Muscle B, attaching 2.5 cm from the joint axis, exerts 260 N of force. (Assume that both muscles attach perpendicular to the bone.) How much torque is created at the joint by each muscle? What is the net torque created at the joint? In which direction will motion at the joint occur? 12. Locate a set of needle-type (or analog) bathroom scales. Assume an erect-standing position on the platform of the scales and have a friend record the weight shown. With this friend kneeling beside the dial of the scales so that he (or she) can see clearly what happens, suddenly relax the muscles of your legs and let your body drop for a short distance. If you do this right, the needle will rapidly swing to a lower reading and then just as rapidly swing back in the opposite direction to a value well above the reading recorded initially. Then, finally, as you come to a complete stop with your legs partly flexed, the needle will oscillate back and forth — because you so violently set it vibrating — and eventually settle at the weight you recorded initially. a. What two vertical forces are acting on your body when you are initially standing on the scales? b. What is the magnitude of the resultant of these two forces when you are standing on the scales? c. What is your vertical acceleration when you are standing on the scales? d. In what direction is your vertical acceleration when you initiate the dropping action? e. In what direction does the resultant of the two forces act when you initiate the dropping motion — upward or downward? f. Do the changes in the reading on the scale indicate that your weight changes during the course of the exercise you were asked to perform? Do you lose weight as you descend and then get it back again later? g. How do you explain the down-up-down sequence in the readings on the scale when you perform a down-and-stop motion?