Moments of Inertia/Center of Mass

[raychelle93]

Could someone help me get started on these two types of problems? (In terms of what equations to use/what the problems are asking for or mean)

Moments of inertia
-A rigid body consists of 2 point masses m1= 1kg at a position vector r1= (1,2,3) m and m2 = 2kg at a position vector r2- (0,1,0) m. Calculate the moments of inertia of this body about the x, y and z axes.

-Four uniform solid spheres of equal mass M = 100 g and radius R = 3 cm are arranged in a square and rigidly connected by four rods of equal mass m = 30 g and length L = 10cm

a) calculate the moments of inertia of the system about the axis AB through the centers of the opposite sides of the square.
b) calculate the moments of inertia of the system about the axis A'B' through the vertices of the square.

(Figure is/looks like a square with A' at the top left, nothing marked at top right, M marked at bottom left and B' marked at bottom right. There is a dotted line in the center of the square marked A B and a diagonal dotted line across the square from points A' to B'. L is from M to B' and 2R is from the top of M to the bottom of M (imagine each vertices to be a sphere, therefore 2R is from the top of the sphere to the bottom).

Center of Mass
-Three point masses 30 g each are placed at the vertices of an equilateral triangle ABC and rigidly connected by three rods of length 10 cm. The Rods AB and AC have equal mass 50 g while the mass of the rod BC is 20 g.

Calculate the distance from the center of mass to the vertex A.

(Figure of Triangle has A vertex at top, C at the bottom left and B at the bottom right)

I have no idea how to get started on the Moments of Inertia problem, in fact if you could explain what exactly the problem is looking for that'd be great; but for the center of mass problem I know the equation is m1x1+m2x2+m3x3/m1+m2+m3 but I don't know what x is and how to use the mass of the rods in the equation...

Again any help would be extremely useful

 

[jbsteele]

Moments of Inertia: The moment of inertia of a rigid body about an axis is a measure of the body's resistance to rotations about that axis. The moment of inertia is calculated by multiplying the mass of each point in the body by the square of its distance from the axis of rotation, then summing up all of these values. For example, for the first problem you provided, the following equations would be used to calculate the moments of inertia about the x, y and z axes: Ix = m1*(1^2 + 2^2 + 3^2) + m2*(0^2 + 1^2 + 0^2) Iy = m1*(1^2 + 0^2 + 3^2) + m2*(2^2 + 1^2 + 0^2) Iz = m1*(1^2 + 2^2 + 0^2) + m2*(0^2 + 1^2 + 3^2) For the second problem, you need to calculate the moments of inertia of the system about the axis AB through the centers of the opposite sides of the square and the axis A'B' through the vertices of the square. To do this, you will need to calculate the position vectors of the four spheres relative to the axes AB and A'B'. Then, you can use the same equation as above to calculate the moments of inertia. Center of Mass: The center of mass of a system of particles is the point around which the entire system's mass is evenly distributed. The distance from the center of mass to any particle can be calculated using the equation x = (m1*x1 + m2*x2 + m3*x3 + ... + mn*xn)/(m1 + m2 + m3 + ... + mn), where m1, m2, m3 etc. are the masses of the particles and x1, x2, x3 etc. are their position vectors relative to the center of mass. For the problem you provided, the position vector of the vertex A relative to the center of mass is what you are looking for, so you can calculate it using the equation above. The position vectors of the other particles (

 

[kerimek]

Sure, I can help you get started on these types of problems! Let's start with moments of inertia.

Moments of inertia are a measure of an object's resistance to rotational motion. In simpler terms, it tells us how difficult it is to make an object rotate. The equation for moment of inertia is:

I = ∑mr^2

Where I is the moment of inertia, m is the mass of each point in the object, and r is the distance from that point to the axis of rotation. The ∑ symbol means to sum up all the individual moments of inertia from each point in the object.

Now, let's take a look at the first problem. We have two point masses, each with a position vector given. To calculate the moment of inertia about a particular axis, we need to find the distance from each point to that axis. For example, to find the moment of inertia about the x-axis, we need to find the distance from each point to the x-axis. We can do this using the Pythagorean theorem.

For the first point, the distance to the x-axis is √(2^2 + 3^2) = √13 m. For the second point, the distance to the x-axis is √(1^2 + 0^2) = 1 m. Now, we can plug these values into the equation for moment of inertia and sum them up to get the total moment of inertia about the x-axis.

I = (1kg)(√13 m)^2 + (2kg)(1m)^2 = 11 kgm^2

You can use a similar method to calculate the moments of inertia about the y and z axes.

For the second problem, we have a more complex system consisting of four spheres and four rods. To calculate the moment of inertia about a particular axis, we will need to use the parallel axis theorem. This theorem states that the moment of inertia about any axis is equal to the moment of inertia about the parallel axis passing through the center of mass plus the product of the total mass and the square of the distance between the two axes.

So for part a, we will need to find the moment of inertia about the axis through the center of mass, which can be found using the equation for a solid sphere:

I = (2/5)MR^2

Where M is the mass of one sphere and R is the radius. Then, we