Center of Mass and Centroid of a Wire
Topics You Need To Understand For This Page |
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how to solve word problems basics of moments and center of mass |
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Interestingly, we can calculate the center of mass and the centroid of a wire.   Here is a video explaining this in detail.
video by Michel vanBiezen |
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For composite wires, the calculations are very similar but instead of areas like we did in the previous section, we use lengths. If you understood those equations, these should be fairly intuitive.
\(\displaystyle{ \bar{x} = \frac{\sum{ x_iL_i }}{\sum{L_i}} }\) |
\(\displaystyle{ \bar{y} = \frac{\sum{ y_iL_i }}{\sum{L_i}} }\) |
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Before we go on, work these practice problems involving the centroids and center of mass of a wire.
Practice
Unless otherwise instructed, solve these problems giving your answers in exact, simplified form.
Find the center of gravity of a semi-circle of radius \(R\) of very thin wire.
Problem Statement |
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Find the center of gravity of a semi-circle of radius \(R\) of very thin wire.
Hint |
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Center the circle at the origin, which, because of symmetry, causes \( \bar{x} = 0 \)
Problem Statement
Find the center of gravity of a semi-circle of radius \(R\) of very thin wire.
Hint
Center the circle at the origin, which, because of symmetry, causes \( \bar{x} = 0 \)
Solution
video by Michel vanBiezen |
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Find the center of gravity of a thin wire shaped in the arc of circle with radius \(R\) and angle \(2\alpha\). Center the circle so that the origin is at the center and the positive x-axis cuts the sector into two equal sections, \(\alpha\) above the x-axis and \(\alpha\) below the x-axis.
Problem Statement
Find the center of gravity of a thin wire shaped in the arc of circle with radius \(R\) and angle \(2\alpha\). Center the circle so that the origin is at the center and the positive x-axis cuts the sector into two equal sections, \(\alpha\) above the x-axis and \(\alpha\) below the x-axis.
Solution
video by Michel vanBiezen |
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Find the center of gravity of a composite wire in the shape of a right triangle where the two legs of the right angle are 10cm and 30cm.
Problem Statement
Find the center of gravity of a composite wire in the shape of a right triangle where the two legs of the right angle are 10cm and 30cm.
Solution
video by Michel vanBiezen |
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Considering the wire as shown in this figure, calculate the force labeled in red in the figure.   We assume that the wire is strong enough to hold the semi-circular shape.
Problem Statement |
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Considering the wire as shown in this figure, calculate the force labeled in red in the figure.   We assume that the wire is strong enough to hold the semi-circular shape.
Hint |
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For a semi-circular wire, \( \bar{x} = 2R/\pi \)
Problem Statement
Considering the wire as shown in this figure, calculate the force labeled in red in the figure.   We assume that the wire is strong enough to hold the semi-circular shape.
Hint
For a semi-circular wire, \( \bar{x} = 2R/\pi \)
Solution
video by Michel vanBiezen |
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