\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Physics - Moments, Center of Mass and Centroids of Composite Shapes

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Moments, Center of Mass and Centroids of Composite Shapes

Centroid of Composite Shapes

Similar to how you found the centroid of individual points, we can find the centroid of composite shapes. The equations are

\(\displaystyle{ \bar{x} = \frac{\sum{ x_iA_i }}{\sum{A_i}} }\)

\(\displaystyle{ \bar{y} = \frac{\sum{ y_iA_i }}{\sum{A_i}} }\)

These equations may look complicated but they are quite understandable once you know what all the symbols mean. Here is a video that goes through an example and explains how this works in detail.

Engineer4Free - How to find the centroid of simple composite shapes [8min-54sec]

video by Engineer4Free

Okay, that ought to be enough to get you started on these practice problems.

Calculus: Early Transcendental Functions 3rd Edition

Practice

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

Find the center of gravity of a composite plate as shown in this figure with units in meters.

Problem Statement

Find the center of gravity of a composite plate as shown in this figure with units in meters.

Solution

Michel vanBiezen - 3578 video solution

video by Michel vanBiezen

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Find the center of gravity of a composite plate as shown in this figure with units in meters.

Problem Statement

Find the center of gravity of a composite plate as shown in this figure with units in meters.

Solution

Michel vanBiezen - 3579 video solution

video by Michel vanBiezen

Log in to rate this practice problem and to see it's current rating.

Find the center of gravity of a composite plate as shown in this figure with units in centimeters.   The full circle on the right is a hole with center at \( (22,0) \)

Problem Statement

Find the center of gravity of a composite plate as shown in this figure with units in centimeters.   The full circle on the right is a hole with center at \( (22,0) \)

Hint

Of course, because of symmetry, \( \bar{y} = 0 \)

Problem Statement

Find the center of gravity of a composite plate as shown in this figure with units in centimeters.   The full circle on the right is a hole with center at \( (22,0) \)

Hint

Of course, because of symmetry, \( \bar{y} = 0 \)

Solution

Michel vanBiezen - 3580 video solution

video by Michel vanBiezen

Log in to rate this practice problem and to see it's current rating.

Find the center of gravity of a composite plate as shown in this figure with units in centimeters.   The full circle is a hole.

Problem Statement

Find the center of gravity of a composite plate as shown in this figure with units in centimeters.   The full circle is a hole.

Solution

Michel vanBiezen - 3581 video solution

video by Michel vanBiezen

Log in to rate this practice problem and to see it's current rating.

Calculate the first moment about the y-axis of the composite plate shown in the figure.

Problem Statement

Calculate the first moment about the y-axis of the composite plate shown in the figure.

Solution

Michel vanBiezen - 3584 video solution

video by Michel vanBiezen

Log in to rate this practice problem and to see it's current rating.

Calculate the first moment about the y-axis of the composite plate shown in the figure.   The circle is a hole.

Problem Statement

Calculate the first moment about the y-axis of the composite plate shown in the figure.   The circle is a hole.

Solution

Michel vanBiezen - 3585 video solution

video by Michel vanBiezen

Log in to rate this practice problem and to see it's current rating.

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

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