Mass, centroids, and moments

A rod on [0,L] with density 1 has mass L. Its first moment about zero is ∫₀ᴸx dx=L²/2, so its center of mass is L/2.

m

2

M

2

x̄

1

For L=4, where is the center of mass?

⏎

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✓The midpoint is L/2=2. Not yet. Use the formula and the values stated in the question, then try again. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

For uniform density 1 under y=x on [0,b], mass is b²/2. My=∫x·x dx=b³/3 and Mx=∫x²/2 dx=b³/6. Hence the centroid is (2b/3,b/3).

m

0.5

x̄

0.666667

ȳ

0.333333

For b=3, what is the centroid’s y-coordinate?

⏎

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✓ȳ=b/3=1. Not yet. Use the formula and the values stated in the question, then try again. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

For the same triangular lamina, Ix=∫₀ᵇx³/3 dx=b⁴/12 and Iy=∫₀ᵇx²·x dx=b⁴/4. These are about the coordinate axes, not axes through the centroid.

Iₓ

0.083333

Iᵧ

0.25

Iᵧ/Iₓ

3

For b>0, what is Iy/Ix?

⏎

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✓The common factor b⁴ cancels: (1/4)/(1/12)=3. Not yet. Use the formula and the values stated in the question, then try again. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.

For a rod with density ρ(x)=x on [0,b], mass is b²/2 and first moment is b³/3. The center of mass 2b/3 lies toward the denser right end, beyond the midpoint.

m

0.5

M

0.333333

x̄

0.666667

For b=3, where is the center of mass?

⏎

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✓2b/3=2, which is right of the midpoint 1.5. Not yet. Use the formula and the values stated in the question, then try again. Enter a finite number or expression, such as 1/2, sqrt(2), or pi. Decimal commas are also accepted.