Chapter 7, Problem 56P

To determine

The position of the center of mass for a high jumper.

To identify: Whether the center of mass lie be outside the body.

Answer to Problem 56P

The center of mass of a bent arm is $9.4\%$ of full-body height below the torso’s median line. Yes, it will lie outside.

Explanation of Solution

Given:

The center of mass of head, neck, and trunk are all lined up on the torso median line.

According to the table, the body followed by height (downwards is the +y direction) and then the mass of body part (in percentage with the whole body mass).

  $\begin{array}{l}\text{On median line: Head }\left(\text{h}\right)\text{: 0 ; 6}\text{.9\% body mass}\hfill \\ \text{ Trunk \& neck }\left(\text{tn}\right)\text{ 0 ; 46}\text{.1\% body mass}\hfill \\ \text{From shoulder hinge point: Upper arms }\left(\text{ua}\right)\text{ 81}\text{.2-71}\text{.7=9}\text{.5 ; 6}\text{.6\% body mass}\hfill \\ \text{ Lower arms }\left(\text{la}\right)\text{ 81}\text{.2-55}\text{.3=25}\text{.9 ; 4}\text{.2\% body mass}\hfill \\ \text{ Hands }\left(\text{ha}\right)\text{ 81}\text{.2-43}\text{.1=38}\text{.1 ; 1}\text{.7\% body mass}\hfill \\ \text{From hip hinge point: Upper legs }\left(\text{ul}\right)\text{ 52}\text{.1-42}\text{.5=9}\text{.6 ; 21}\text{.5\% body mass}\hfill \\ \text{ Lower legs }\left(\text{ll}\right)\text{ 52}\text{.1-18}\text{.2=33}\text{.9 ; 9}\text{.6\% body mass}\hfill \\ \text{ Feet }\left(\text{f}\right)\text{ 52}\text{.1-1}\text{.8=50}\text{.3 ; 3}\text{.4\% body mass}\hfill \end{array}$

  

Formula used:

The center of mass is calculated as,

  $y_{cm}=\frac{{\displaystyle \sum my}}{{\displaystyle \sum m}}$

Calculation:

Considering the date from the table as shown above, the vertical location of the center of mass is

  $\begin{array}{c}{y}_{cm}=\frac{y_h{m}_h+y_{tn}{m}_{tn}+y_{ua}{m}_{ua}+y_{la}{m}_{la}+y_{ha}{m}_{ha}+y_{ul}{m}_{ul}+y_{ll}{m}_{ll}+y_f{m}_f}{m_{full body}}\\ =\frac{0+0+(9.5)(6.6)+(25.9)(4.2)+(38.1)(1.7)+(9.6)(21.5)+(33.9)(9.6)+(50.3)(3.4)}{100}\\ =9.4\end{array}$

The center of mass of a bent arm is $9.4\%$ of full-body height below the torso’s median line. For a person of height 1.6 m, this is about 15 cm, which is slightly outside the body.