Figure 10–72 illustrates an H₂O molecule. The O―H b...

Question

Figure 10–72 illustrates an H₂O molecule. The O―H bond length is 0.096 nm and the H―O―H bonds make an angle of 104°. Calculate the moment of inertia of the H₂O molecule (assume the atoms are points) about an axis passing through the center of the oxygen atom perpendicular to the plane of the molecule.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice prom together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Three metal spheres that each weigh 1 kg are connected by rigid rods of negligible masses. As shown in the figure below the distance between two adjacent spheres is 0.40 m. If the spheres at the edges make an angle of 180 degrees with the one in the middle, determine what the moment of inertia for this particular system of metal spheres about an axis perpendicular to the plane of the spears going through the center of the sphere in the middle will be hint Trent the spears as points. Awesome. So our end goal is ultimately, we're trying to figure out what the moment of inertia is for this particular system of metal spears, considering that it's about the axis perpendicular to the plane of the spears going through the center of the sphere in the mental in the middle. And we're also asked to treat the metal spheres like points. Awesome. So let's quickly look at our diagram here that's given to us by the problem itself. And as we could see, we have three gray circles denoting our three metal spheres that are each 1 kg each and the distance between two spheres. So from going from the top to the bottom, so from the first sphere to the second sphere is 0.40 m. And from the second sphere to the third sphere, it's 0.40 m and it forms an angle of 180 degrees as shown by this semi circle shape, which is there's a black line connecting the first sphere to the second sphere and a black line connecting the second sphere to the third sphere. So in between, we have a black cir a black semi circle curved line which is connecting between the 1st and 2nd sphere to the 2nd and 3rd sphere which is 180 degrees. Awesome. And so now that we have a visualization of what's going on here and noting that our 0.40 m, the distance between the spheres are denoted by green arrows and green lines. Now that we have a visualization of what's happening here, let's look at our multiple choice answers to see what our final answer might be. And let's note that all of our multiple choice answers are in units of kilograms multiplied by meters squared. So A is zero B is 0.32 C is 0.48 and D is 1.3. Awesome. So with that in mind, first off, let us note that since the axis passes through the sphere in the moment and the my sorry in the middle. So since this, you know, the axis passes through the sphere in the middle, we can recall and write the moment of inertia equation as one half multiplied by M multiplied by R squared where R is the radius of the sphere and M is the mass. So plugging in our value for the radius, so one half multiplied by M multiplied by, since we're not given a radius value, we can just say the radius is zero. So this is all equal to zero. Awesome. So let us also note that the moment of inertia for the other two spheres about the axis will be equal since they are of equal size and are at equal distances. So let us note that we are given that the mass of the sphere which I'm gonna call M is equal to 1 kg and the distance between the two adjacent spheres, let's call this D is equal to 0.40 m. And once again, this is the distance between the two adjacent spheres. So now we can plug in all of our known variables into the total mo total moment of inertia equation to solve for our final answer. Let us recall that for the total moment of inertia, I is equal to two multiplied by M multiplied by D squared. Awesome. So plugging in our known values to solve for I I is equal to two multiplied by 1 kg. Awesome multiplied by our D value which is 0.40 m squared. So when we plug that into our calculator, we will find that I, which is our total moment of inertia is equal to 0.32 and its units are kilograms multiplied by meters squared. And that's it we solve for this problem. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter B 0.32 kg multiplied by meters squared. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Moment of Inertia

Molecular Geometry

Coordinate System

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