In the ammonia \(\left(\mathrm{NH}_{3}\right)\) molecule, the three hydrogen (H) atoms form an equilateral triangle, the distance between centers of the atoms being \(16.28 \times 10^{-11} \mathrm{~m}\), so that the center of the triangle is \(9.40 \times 10^{-11} \mathrm{~m}\) from each hydrogen atom. The nitrogen \((\mathrm{N})\) atom is at the apex of a pyramid, the three hydrogens constituting the base (see Fig. 7-27). The nitrogen/hydrogen distance is \(10.14 \times 10^{-11} \mathrm{~m}\) and the nitrogen/hydrogen atomic mass ratio is \(13.9 .\) Locate the center of mass relative to the nitrogen atom.

The problem has been set so the origin lies at the nitrogen atom, which simplifies matters. The system is symmetric in the plane of the hydrogen atoms, and by symmetry the center of mass must be on the line (z-direction) drawn from the nitrogen atom through the center of the equilateral triangle.

The center of mass can be described mathematically as follows: \[z_{C M}=\frac{\sum_{i} m_{i} z_{i}}{\sum_{i} m_{i}}\]Here \(m_{i}\) and \(z_{i}\) are respective masses and z coordinates. For this problem, \(z_{C M}\) will be the desired distance from the nitrogen atom (which is at the origin) to the center of mass. The sum goes over all 4 atoms - one nitrogen and three hydrogens. We assume that the entire mass of each atom is concentrated at it's center, allowing us to ignore size. The z position \(z_{i}\) is negative for nitrogen and positive for each hydrogen. The z coordinate is zero for the center of the base, thus it is \(9.4 \times 10^{-11}\) m for each hydrogen atom. By plugging values into the equation, the following is obtained: \[z_{C M}=\frac{(-14.003 \times 10^{-27} \mathrm{kg}(-10.14 \times 10^{-11} \mathrm{m}))+3(1.007 \times 10^{-27} \mathrm{kg}(9.40 \times 10^{-11} \mathrm{m}))}{-14.003 \times 10^{-27} \mathrm{kg}+3(1.007 \times 10^{-27} \mathrm{kg})}\]

After solving the equation from step 2, we find that \(z_{C M}\) is equal to \(0.56 \times 10^{-11} \mathrm{m}\), which is the distance upward from the nitrogen atom (towards the plane of hydrogen atoms). That is, the center of mass is slightly on the nitrogen side of the geometric center (halfway up). This is expected, because the nitrogen atom is more massive.