A molecule consists of identical atoms of mass m placed at each vertex of a regular hexagon of side a. Calculate the moment of inertia of the molecule about (a) the z-axis that is perpendicular to the plane of the hexagon and passing through its center, and (b) the y-axis that passes through two diametrically opposite atoms.
The Correct Answer and Explanation is :
Let’s break this problem into parts and calculate the moment of inertia for both axes.
Part (a): Moment of inertia about the z-axis
- Shape of the molecule: The molecule consists of 6 identical atoms placed at the vertices of a regular hexagon. Let the mass of each atom be mm, and the distance between adjacent atoms be aa. The hexagon is symmetrical, so we can assume that the atoms are positioned in a plane (xy-plane).
- Moment of inertia about the z-axis: The z-axis is perpendicular to the plane of the hexagon and passes through its center. To find the moment of inertia, we use the formula for the moment of inertia of a point mass: I=∑miri2I = \sum m_i r_i^2 where mim_i is the mass of the atom and rir_i is the perpendicular distance of the atom from the axis of rotation. In this case, since the atoms are placed at the vertices of a regular hexagon, the distance from each atom to the center of the hexagon (the z-axis) is the same for all atoms. The distance from the center of the hexagon to any vertex (which forms an equilateral triangle with the center) is given by: r=a3r = \frac{a}{\sqrt{3}} where aa is the side of the hexagon.
- Calculating the moment of inertia: Since there are 6 atoms, each with mass mm, the total moment of inertia about the z-axis is: Iz=6⋅m⋅r2=6⋅m⋅(a3)2=6⋅m⋅a23=2ma2I_z = 6 \cdot m \cdot r^2 = 6 \cdot m \cdot \left(\frac{a}{\sqrt{3}}\right)^2 = 6 \cdot m \cdot \frac{a^2}{3} = 2ma^2 So, the moment of inertia about the z-axis is: Iz=2ma2I_z = 2ma^2
Part (b): Moment of inertia about the y-axis passing through two diametrically opposite atoms
- Coordinates of the atoms: Let’s place the center of the hexagon at the origin, and align the hexagon such that two diametrically opposite atoms lie along the x-axis. For a regular hexagon, the coordinates of the atoms can be written as: (xi,yi)=(rcos(θi),rsin(θi))(x_i, y_i) = (r \cos(\theta_i), r \sin(\theta_i)) where r=a3r = \frac{a}{\sqrt{3}} and θi=πi3\theta_i = \frac{\pi i}{3} for i=0,1,2,…,5i = 0, 1, 2, \dots, 5.
- Moment of inertia about the y-axis: The y-axis passes through two diametrically opposite atoms, so we need to compute the moment of inertia for all the atoms about this axis. For each atom, the moment of inertia about the y-axis is given by: Iy=∑mixi2I_y = \sum m_i x_i^2 where xix_i is the horizontal distance of each atom from the y-axis. Since the y-axis passes through two opposite atoms, the total contribution of those two atoms will be zero, as their xix_i coordinates are zero. The moment of inertia for the other 4 atoms is given by the sum of their xi2x_i^2 terms. Since the hexagon is symmetric, we can compute the moment of inertia for one atom and then multiply by 4. For a single atom at (rcos(θ),rsin(θ))(r \cos(\theta), r \sin(\theta)), the distance from the y-axis is x=rcos(θ)x = r \cos(\theta). Therefore: Iy=4⋅m⋅r2=4⋅m⋅(a3)2=4⋅m⋅a23=4ma23I_y = 4 \cdot m \cdot r^2 = 4 \cdot m \cdot \left(\frac{a}{\sqrt{3}}\right)^2 = 4 \cdot m \cdot \frac{a^2}{3} = \frac{4ma^2}{3}
Final answers:
- Moment of inertia about the z-axis: Iz=2ma2I_z = 2ma^2
- Moment of inertia about the y-axis: Iy=4ma23I_y = \frac{4ma^2}{3}
Explanation:
- The moment of inertia about the z-axis is computed based on the fact that all the atoms are equidistant from the center of the hexagon, and the z-axis is perpendicular to the plane of the hexagon. Thus, the distance rr from the center to each vertex is the same, and we use the formula I=∑mir2I = \sum m_i r^2 for each atom.
- The moment of inertia about the y-axis is computed by considering the projection of each atom’s position along the axis. For this, we recognize that two atoms lie along the y-axis, contributing zero to the moment of inertia. The other atoms contribute based on their distance from the y-axis, which is determined by their x-coordinates.
The symmetry of the hexagon and the distribution of the atoms simplifies the calculation and allows us to use the known geometric properties of regular polygons.