Three cubes, of side 0, 20, and 30, are placed next to one another (in contact) with their centers along a straight line as shown in the figure. What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform material and 0 = 5.5 cm.
A cube with side dimension of zero does not exist.
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Three cubes, of side Lₒ, 2Lₒ, and 3Lₒ, are placed next to one another (in contact) with their centers along a straight line as shown in the figure. What is the position, along this line, of the CM of this system? Assume the cubes are made of the same uniform material and Lₒ = 5.5 cm.
To find the position of the center of mass (CM) of the system, we need to consider the mass and the position of each cube.
First, let's calculate the mass of each cube. We'll assume that the density of the material is uniform.
The volume of a cube is given by V = side^3. Therefore, the volume of the first cube (side length of 0) is 0^3 = 0 cubic cm. Since it has zero volume, its mass is zero as well.
The volume of the second cube (side length of 20 cm) is 20^3 = 8000 cubic cm. Since the volume of each cube is directly proportional to its side length cubed, we can say that the mass of the second cube is directly proportional to the side length cubed as well. Therefore, the mass of the second cube is (20/5.5)^3 = (20/5.5)*(20/5.5)*(20/5.5) = 102.857 grams.
Similarly, the volume of the third cube (side length of 30 cm) is 30^3 = 27000 cubic cm. Therefore, the mass of the third cube is (30/5.5)^3 = (30/5.5)*(30/5.5)*(30/5.5) = 915.755 grams.
Now, let's calculate the position of the CM. Since the first cube has no mass, it won't affect the position of the CM. We only need to consider the masses and positions of the second and third cubes.
Let's assume the position of the CM is measured from the left end of the line. The second cube is located at a distance of 20/2 = 10 cm from the left end, and the third cube is located at a distance of 20 + 30/2 = 35 cm from the left end.
To find the position of the CM, we'll use the principle of moments. The principle of moments states that for a system to be balanced, the sum of the anticlockwise moments must be equal to the sum of the clockwise moments.
The moment of an object is given by the product of its mass and its distance from the reference point. So, the moment of the second cube is given by M2 = mass2 * distance2 = 102.857 grams * 10 cm = 1028.57 gram-cm, and the moment of the third cube is given by M3 = mass3 * distance3 = 915.755 grams * 35 cm = 32045.43 gram-cm.
For the system to be balanced, the sum of the anticlockwise moments (M2 and M3) must be equal to the sum of the clockwise moments (which is zero in this case since there is only one cube on the left side).
Therefore, M2 + M3 = 0
1028.57 gram-cm + 32045.43 gram-cm = 0
Total moment = -33074 gram-cm
Now, let's find the position of the CM, which we'll call X.
The formula for the position of the CM is given by X = total moment / total mass.
Here, the total mass is the sum of the masses of the second and third cubes, which is 102.857 + 915.755 grams = 1018.612 grams.
Therefore, X = (-33074 gram-cm) / (1018.612 grams) = -32.45 cm.
Since the position of the CM is measured from the left end of the line, the negative sign indicates that the CM is located 32.45 cm to the left of the left end of the line.
Therefore, the position of the CM is -32.45 cm.