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17Calculus Physics - Center of Mass and Centroids of a Wire

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Center of Mass and Centroid of a Wire

Interestingly, we can calculate the center of mass and the centroid of a wire.   Here is a video explaining this in detail.

Michel vanBiezen - Mechanical Engineering: Centroids & Center of Gravity - Center of Gravity of a Wire

video by Michel vanBiezen

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.

\(\displaystyle{ \bar{x} = \frac{\sum{ x_iL_i }}{\sum{L_i}} }\)

\(\displaystyle{ \bar{y} = \frac{\sum{ y_iL_i }}{\sum{L_i}} }\)

Before we go on, work these practice problems involving the centroids and center of mass of a wire.

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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

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 \( \bar{x} = 0 \)

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

Michel vanBiezen - 3576 video 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

Michel vanBiezen - 3577 video 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

Michel vanBiezen - 3582 video 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

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, \( \bar{x} = 2R/\pi \)

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

Michel vanBiezen - 3583 video solution

video by Michel vanBiezen

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Practice Instructions

Unless otherwise instructed, solve these problems giving your answers in exact, simplified form.

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