Verify that the force in the elbow joint inExample\({\rm{9}}{\rm{.4}}\)is \({\rm{407}}\;{\rm{N}}\), as stated in the text.

A force is an external factor that can change the rest or motion of a body. It has a size and a general direction.

The diagram is given below:

The torque around a point of equilibrium is zero. So, we write,

\(\begin{align}{F_E}{r_1} &= {w_a}\left( {{r_2} - {r_1}} \right) + {w_b}\left( {{r_3} - {r_1}} \right)\\{F_E} &= \frac{{{w_a}\left( {{r_2} - {r_1}} \right) + {w_b}\left( {{r_3} - {r_1}} \right)}}{{{r_1}}}\\{F_E} &= {w_a}\left( {\frac{{{r_2}}}{{{r_1}}} - 1} \right) + {w_b}\left( {\frac{{{r_3}}}{{{r_1}}} - 1} \right)\end{align}\)

Here, the forearm's weight is,

\(\begin{align}{w_a} &= 2.5 \times 9.8\\ &= 24.5\;N\end{align}\)

The load's total weight is,

\(\begin{align}{w_b} &= 4 \times 9.8\\ &= 39.2\;N\end{align}\)

The distance between the biceps and the elbow where they exert force is\({r_1} = 4\;{\rm{cm}}\).

The force exerted by the forearm's weight on the elbow is\({r_2} = 16\;{\rm{cm}}\).

The load and elbow are separated by a distance of\({r_3} = 38\;{\rm{cm}}\).

For all of these reasons,

\(\begin{align}{F_E} &= 24.5 \times \left( {\frac{{16}}{4} - 1} \right) + 39.2 \times \left( {\frac{{38}}{4} - 1} \right)\\ \approx 407\;N\end{align}\)

Hence, it is proved.