Question

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.

Short answer

The center of mass is 4.3% of the full body height, below the torso’s medium line. It is not outside the body.

Step-by-step solution

Define the center of mass (CM)

The CM of a two-particle system lies on the line joining the particles. It is closer to the heavier object. It is the point at which the whole mass of a body is concentrated.

Given information

From table 7-1,

The percent mass of the upper arm is \({m_{{\rm{UA}}}} = 6.6\;{\rm{units}}\).

The percent mass of the lower arm is \({m_{{\rm{LA}}}} = 4.2\;{\rm{units}}\).

The percent mass of the hand is \({m_{\rm{H}}} = 1.7\;{\rm{units}}\).

The percent mass of the lower leg is \({m_{{\rm{LL}}}} = 9.6\;{\rm{units}}\).

The percent mass of feet is \({m_{\rm{F}}} = 3.4\;{\rm{units}}\).

The percent mass of the full body is \({m_{{\rm{Full}}\;{\rm{body}}}} = 100\;{\rm{units}}\).

The distance of the upper arm from the center of mass is \({y_{{\rm{UA}}}} = \left( {81.2 - 71.7} \right)\;{\rm{units}}\).

The distance of the lower arm from the center of mass is \({y_{{\rm{LA}}}} = \left( {81.2 - 55.3} \right)\;{\rm{units}}\).

The distance of the hand from the center of mass is \({y_{\rm{H}}} = \left( {81.2 - 43.1} \right)\;{\rm{units}}\).

The distance of the lower leg from the center of mass is \({y_{{\rm{LL}}}} = \left( {28.5 - 18.2} \right)\;{\rm{units}}\).

The distance of the foot from the center of mass is \({y_{\rm{F}}} = \left( {28.5 - 1.8} \right)\;{\rm{units}}\).

Calculate the center of mass

The center of mass can be calculated as shown below:

\(\begin{array}{l}{y_{{\rm{cm}}}} = \frac{{{m_{{\rm{UA}}}}{y_{{\rm{UA}}}} + {m_{{\rm{LA}}}}{y_{{\rm{LA}}}} + {m_{\rm{H}}}{y_{\rm{H}}} + {m_{{\rm{LL}}}}{y_{{\rm{LL}}}} + {m_{\rm{F}}}{y_{\rm{F}}}}}{{{m_{{\rm{Full}}\;{\rm{body}}}}}}\\{y_{{\rm{cm}}}} = \frac{{\left[ \begin{array}{l}\left( {6.6} \right)\left( {81.2 - 71.7} \right) + \left( {4.2} \right)\left( {81.2 - 55.3} \right) + \left( {1.7} \right)\left( {81.2 - 43.1} \right)\\ + \left( {9.6} \right)\left( {28.5 - 18.2} \right) + \left( {3.4} \right)\left( {28.5 - 1.8} \right)\end{array} \right]}}{{\left( {100} \right)}}\\{y_{{\rm{cm}}}} = 4.25 \approx 4.3\end{array}\)

The center of mass is 4.3% of the full body height, below the torso’s medium line.

For a person of height 1.7 m, this will be

\(\begin{array}{l}{y_{{\rm{cm}}}} = \left( {4.3\% } \right)\left( {1.7\;{\rm{m}}} \right)\left( {\frac{{100\;{\rm{cm}}}}{{1\;{\rm{m}}}}} \right)\\{y_{{\rm{cm}}}} = 7.24\;{\rm{cm}}{\rm{.}}\end{array}\)

The thickness of the torso from the median line is about 10 cm. Therefore, the CM is well inside the body and not outside it.