When a high jumper is in a position such that his arms and lower legs are hanging vertically, and his thighs, trunk, and head are horizontal just above the bar, estimate how far below the torso’s median line the CM will be. Will this CM be outside the body? Use Table 7–1.

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. We are going to find the y-coordinate of the center of mass of this high jumper when their position with their upper legs and torso and head all horizontal and their lower legs and feet and upper arm, lower arm and hand all below the mid line hanging down like this. So we'll multiply the mass of each portion of their body by the position of the center of mass of that part and add the products together and divide by the mass of the entire body to get the position of the center of mass. So for the lower leg, its center of mass is here at a height 18.2 percent of their total height above the ground and we'll take the position of the knee which is 28.5 percent of the total height above the ground and that difference is gonna be 10.3 and so that's the distance below the knee to the center of mass of the lower leg, 10.3 there. And it's the same idea for the foot; it's the knee position minus the center of mass of the foot position and that gives 26.7 percent of the person's height is the distance below the knee to the center of mass of the foot, 26.7. And the same idea for the arm and hand except the shoulder is the reference point now so the shoulder is 81.2 percent of the person's height above the ground and the center of mass of the upper arm is 71.7 percent of the person's height above the ground and you subtract the two and you get 9.5 will be the distance from the shoulder joint to the center of mass of the upper arm and then same idea for the lower arm, 25.9 and then for the hand, you get 38.1. And so you multiply these positions for the lower arm by the mass of the lower arm and then the y-position of the center of the foot with respect to the median line or the knee multiply by the mass of the foot and then the y-coordinate of the center of mass of the upper arm with respect to the median or the shoulder, in other words, and then so on and so on for the lower arm and the hand and we get all this. So 9.6 of the person's total mass is contained within the lower leg and then multiply that by the 10.3 position of the center of mass of the lower leg and then the foot accounts for 3.4 percent of a person's mass times by 26.7 and then the upper arm is 6.6 times 9.5 lower arm is 4.2 percent of the mass times 25.9 and then the hand is only 1.7 but it has a compensating large number for it's hanging down a long way so even though it's a small mass, it contributes somewhat significantly to moving the center of mass downwards because its distance from the shoulder or from the body's median line is so far, 38.1. And the calculator looks like this; make sure you have brackets there around the numerator and you get 4.3 percent below the torso median line will be the y-coordinate of the person's center of mass. For a person who's maybe 155 centimeters tall, this would be, let's see, about 6.6 centimeters which is not very much; 6.6 centimeters below the middle of their body and that's probably inside the body still so actually I should put this center of mass maybe around here somewhere basically just before their shirt is gonna be the position of their center of mass.