VECTOR MECHANICS FOR ENGINEERS: STATICS
VECTOR MECHANICS FOR ENGINEERS: STATICS
12th Edition
ISBN: 9781260912814
Author: BEER
Publisher: MCG
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Chapter 5.1, Problem 5.16P

Chapter 5.1, Problem 5.16P, PROBLEM 5.16 Determine the y coordinate of the centroid of the shaded area in terms of r1, r2, and .

PROBLEM 5.16

Determine the y coordinate of the centroid of the shaded area in terms of r1, r2, and α.

Expert Solution & Answer
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To determine

The y coordinate of the centroid of the shaded area in terms of r1,r2 and α.

Answer to Problem 5.16P

The y-coordinate of the centroid of the plane area is (23)(r23r13)(r22r12)(cosα90°α)_.

Explanation of Solution

Write the expression for the y-coordinate of the centroid of the sector with radius r2.

y2¯=(23)r2(sin(90°α)90°α)=(23)r2(cosα90°α) (I)

Here y2¯ is the y-coordinate of the centroid of the sector with radius r2.

Write the expression for the area of the sector with radius r2.

A=(90°α)r22 (II)

A is the radius of the sector with radius r2.

Write the expression for the y-coordinate of the centroid of the sector with radius r1.

y1¯=(23)r1(sin(90°α)90°α)=(23)r1(cosα90°α) (III)

Here y1¯ is the y-coordinate of the centroid of the sector with radius r1.

Write the expression for the area of the sector with radius r1.

A=(90°α)r12 (IV)

A is the radius of the sector with radius r1.

Write the expression for y¯A.

y¯A=(y2¯)A(y1¯)A

Substitute (I), (II), (III) and (IV) in the above equation to calculate y¯A.

y¯A=((23)r2(cosα90°α))(90°α)r22((23)r1(cosα90°α))(90°α)r12=(23)(r23r13)cosα (V)

Write the expression for A.

A=AA

Substitute (II) and (IV) in the above equation to calculate A.

A=(90°α)r22(90°α)r12=(90°α)(r22r12) (VI)

Write the expression to calculate the y-coordinate of the centroid of the given area.

Y¯A=y¯A

Y¯ is the y-coordinate of the centroid of the given area.

Conclusion:

Substitute (V) and (VI) in the above equation to calculate Y¯.

Y¯((90°α)(r22r12))=(23)(r23r13)cosαY¯=(23)(r23r13)(r22r12)(cosα90°α)

Thus, the y-coordinate of the centroid of the plane area is (23)(r23r13)(r22r12)(cosα90°α)_.

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Chapter 5 Solutions

VECTOR MECHANICS FOR ENGINEERS: STATICS

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