Ch. 09 - Linear Momentum

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 09 - Linear Momentum

Problem 58

Chapter 9, Problem 58

Find the center of mass of the ammonia molecule. The chemical formula is NH₃. The hydrogens are at the corners of an equilateral triangle (with sides 0.16 nm) that forms the base of a pyramid, with nitrogen at the apex (0.037 nm vertically above the plane of the triangle).

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Identify the positions of the atoms in the ammonia molecule. The nitrogen atom is at the apex of the pyramid, 0.037 nm above the plane of the equilateral triangle formed by the three hydrogen atoms. The hydrogen atoms are at the corners of the triangle, each separated by 0.16 nm.

Assign a coordinate system. Place the center of the equilateral triangle (the centroid of the hydrogen atoms) at the origin of the x-y plane. The nitrogen atom will then have coordinates (0, 0, 0.037 nm), and the hydrogen atoms will have coordinates in the x-y plane based on the geometry of the triangle.

Calculate the center of mass in the x and y directions. Since the hydrogen atoms are symmetrically arranged in the x-y plane, their contributions to the x and y coordinates of the center of mass will cancel out, resulting in the center of mass being at (0, 0) in the x-y plane.

Calculate the center of mass in the z direction. Use the formula for the center of mass along the z-axis:

z

_cm = (Σm_iz_i)/(Σm_i), where m_i is the mass of each atom and z_i is its z-coordinate. Substitute the masses of nitrogen and hydrogen, and their respective z-coordinates (0.037 nm for nitrogen and 0 nm for the hydrogens).

Simplify the expression for z_cm to find the vertical position of the center of mass. Combine the contributions of the nitrogen and hydrogen atoms, taking into account their relative masses (m_N ≈ 14u and m_H ≈ 1u).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Center of Mass

The center of mass is a point that represents the average position of the mass distribution of an object or system. It is calculated by taking into account the mass and position of each constituent particle. In molecular structures, the center of mass can be determined by considering the geometric arrangement of atoms and their respective masses.

Molecular Geometry

Molecular geometry refers to the three-dimensional arrangement of atoms within a molecule. In the case of ammonia (NH₃), the nitrogen atom is at the apex of a pyramid, with the hydrogen atoms forming the base. Understanding the geometry is crucial for accurately calculating the center of mass, as it influences the distances and angles between the atoms.

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are of equal length and all three angles are 60 degrees. In the context of the ammonia molecule, the hydrogen atoms are positioned at the vertices of an equilateral triangle, which simplifies the calculation of the center of mass by providing symmetry in the mass distribution.

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