VECTOR MECHANICS FOR ENGINEERS W/CON >B
VECTOR MECHANICS FOR ENGINEERS W/CON >B
12th Edition
ISBN: 9781260804638
Author: BEER
Publisher: MCG
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Chapter 5.4, Problem 5.123P
To determine

To determine the value of x¯ for the two volumes.

Expert Solution & Answer
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Answer to Problem 5.123P

The value of x¯ for the first component is 2188h, and the value of x¯ for the second component is 2740h.

Explanation of Solution

Refer Fig. P5.21 and Fig. 1.

VECTOR MECHANICS FOR ENGINEERS W/CON >B, Chapter 5.4, Problem 5.123P , additional homework tip  1

Let the element of volume of the disk be of radius r and thickness dx.

Find the equation for radius from the equation of the generating curve.

x2h2+y2a2=1r2=a2h2(h2x2)

Here, the y component is equal to the radius, h is the radius of the semi ellipsoid, x is the centroid of the element and dx is the thickness of the element.

Write the expression for the volume of the element.

dV=πr2dx=πa2h2(h2x2)dx (I)

Find the centroid of the element.

xEL¯=x

Here, xEL¯ is the centroid of the element.

Write the expression for the x¯ for the volume.

x¯V=xEL¯dVx¯=xEL¯dVV (II)

Conclusion:

Refer Fig.2

VECTOR MECHANICS FOR ENGINEERS W/CON >B, Chapter 5.4, Problem 5.123P , additional homework tip  2

Calculate the volume of component 1.

V1=0h/2dV=0h/2πa2h2(h2x2)dx=π(a2h2)[h2xx33]0h/2=1124πa2h

Calculate 1xEL¯dV for component 1.

1xEL¯dV=0h/2x[πa2h2(h2x2)dx]=π(a2h2)[h2x22x44]0h/2=764πa2h2

Substitute 764πa2h2 for 1xEL¯dV and 1124πa2h for V1 in equation (II).

x1¯=764πa2h21124πa2h=2188h

Thus, the value of x¯ for the first component is 2188h.

Calculate the volume of component 2.

V2=h/2hdV=h/2hπa2h2(h2x2)dx=π(a2h2)[h2xx33]h/2h=π(a2h2){[h2(h)h33][h2(h2)(h2)33]}=524πa2h

Calculate 2xEL¯dV for component 2.

2xEL¯dV=a/2ax[πa2h2(h2x2)dx]=π(a2h2)[h2x22x44]h/2h=π(a2h2){[h2h22h44][h2(h2)22(h2)44]}=964πa2h2

Substitute 964πa2h2 for 2xEL¯dV and 524πa2h for V2 in equation (II).

x2¯=964πa2h2524πa2h=2740h

Thus, the value of x¯ for the second component is 2740h.

Therefore, the value of x¯ for the first component is 2188h, and the value of x¯ for the second component is 2740h.

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Chapter 5 Solutions

VECTOR MECHANICS FOR ENGINEERS W/CON >B

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