364

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61. Determine the area and the centroid of the area.(x, y) y

1 m

1 m

y2 x3

x

365

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62. Determine the area and the centroid of the area.(x, y) y

x

3 ft

3 ft y x3

19

366

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63. Locate the centroid of the area.x y

x

2 ft

x1/2 2x5/3y

367

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*64. Locate the centroid of the area.y y

x

2 ft

x1/2 2x5/3y

368

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65. Determine the area and the centroid of the area.(x, y) y

x

a

b

xy c2

369

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66. Determine the area and the centroid of the area.(x, y) y

x

a

h y x2 ha2

370

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67. Locate the centroid ( , ) of the area.yx y

x2 m

1 m

y 1 x214

371

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y

x

2 in.

2 in.

y 1

0.5 in.

0.5 in.

x

*68. Locate the centroid of the area.x

69. Locate the centroid of the area.y y

x

2 in.

2 in.

y 1

0.5 in.

0.5 in.

x

372

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y

x

9 ft

3 ft

y 9 x2

610. Locate the centroid ( , ) of the area.yx

373

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611. Determine the area and the centroid of thearea.

(x, y) y

x

y

y x

3 ft

3 ft

x39

374

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*612. Locate the centroid of the area.x y

x1 m

y x2

1 m

y2 x

613. Locate the centroid of the area.y y

x1 m

y x2

1 m

y2 x

375

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614. Locate the center of mass of the circular coneformed by revolving the shaded area about the y axis. Thedensity at any point in the cone is defined by ,where is a constant.r0

r = (r0 >h)y

y

y

x

z

h

a

z y aah

376

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615. Locate the centroid of the homogeneous solidformed by revolving the shaded area about the y axis.

y

y

x

z

y2 (z a)2 a2

a

377

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*616. Locate the centroid of the solid.z

y

z

x

a

z a1

a a y( )2

378

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617. Locate the centroid of the homogeneous solidformed by revolving the shaded area about the y axis.

y z

y

x

z2 y3116

2 m

4 m

379

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618. Locate the centroid of the homogeneous solidfrustum of the paraboloid formed by revolving the shadedarea about the z axis.

z

a

z (a2 y2)ha2

h2

h2

z

x

y

380

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619. Locate the centroid of the cross-sectional area ofthe concrete beam.

y

x

y

3 in.

6 in.

3 in.

27 in.

3 in.

12 in. 12 in.

381

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*620. Locate the centroid of the cross-sectional area ofthe built-up beam.

y y

x

6 in.1 in.

1 in.

1 in.1 in.

3 in.3 in.

6 in.

382

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622. Locate the distance to the centroid of themembers cross-sectional area.

y

x

y

0.5 in.

6 in.

0.5 in.

1.5 in.

1 in.

3 in. 3 in.

621. Locate the centroid of the channels cross-sectional area.

y

2 in.

4 in.

2 in.12 in.

2 in.

C

y

383

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623. Locate the centroid of the cross-sectional area ofthe built-up beam.

y y

x

1.5 in.

1.5 in.

11.5 in.

1.5 in.

3.5 in.

4in. 1.5 in.4 in.

384

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*624. The gravity wall is made of concrete. Determine thelocation ( , ) of the center of mass G for the wall.yx

y

1.2 m

x

_x

_y

0.6 m 0.6 m2.4 m

3 mG

0.4 m

385

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625. Locate the centroid of the cross-sectional area ofthe built-up beam.

y y

x

450 mm

150 mm150 mm

200 mm

20 mm

20 mm

386

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626. Locate the centroid of the cross-sectional area ofthe built-up beam.

y

200 mm

20 mm50 mm

150 mm

y

x

200 mm

300 mm

10 mm

20 mm 20 mm

10 mm

387

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627. Locate the center of mass for the compressorassembly.The locations of the centers of mass of the variouscomponents and their masses are indicated and tabulated inthe figure.What are the vertical reactions at blocks A and Bneeded to support the platform?

x

y

1

2

34

Instrument panel

Filter system

Piping assembly

Liquid storage

Structural framework

230 kg

183 kg

120 kg

85 kg

468 kg

1

2

34

5

5

2.30 m1.80 m

3.15 m

4.83 m

3.26 m

A B

2.42 m 2.87 m1.64 m1.19m

1.20 m

3.68 m

388

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*628. Major floor loadings in a shop are caused by theweights of the objects shown. Each force acts through itsrespective center of gravity G. Locate the center of gravity( , ) of all these components.yx

z

y

G2

G4G3

G1

x

600 lb9 ft

7 ft

12 ft

6 ft

8 ft4 ft 3 ft

5 ft

1500 lb

450 lb

280 lb

389

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629. Locate the center of mass of thehomogeneous block assembly.

(x, y, z)

y

z

x 150 mm

250 mm

200 mm

150 mm150 mm100 mm

390

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630. Locate the center of mass of the assembly. Thehemisphere and the cone are made from materials havingdensities of and , respectively.4 Mg>m38 Mg>m3

z

y

z

x

100 mm 300 mm

391

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631. Replace the distributed loading with an equivalentresultant force, and specify its location on the beammeasured from point A.

A

B

3 m 3 m

15 kN/m

10 kN/m

3 m

392

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*632. Replace the distributed loading with an equivalentresultant force, and specify its location on the beammeasured from point A.

B

A

8 kN/m

4 kN/m

3 m 3 m

393

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633. Replace the distributed loading by an equivalentresultant force and specify its location, measured frompoint A.

3 m2 m

A B

800 N/m

200 N/m

634. Replace the distributed loading with an equivalentresultant force, and specify its location on the beammeasured from point A.

A B

L2

L2

w0 w0

394

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635. The distribution of soil loading on the bottom ofa building slab is shown. Replace this loading by anequivalent resultant force and specify its location,measured from point O.

12 ft 9 ft

100 lb/ft50 lb/ft

300 lb/ft

O

*636. Determine the intensities and of thedistributed loading acting on the bottom of the slab so thatthis loading has an equivalent resultant force that is equalbut opposite to the resultant of the distributed loadingacting on the top of the plate.

w2w1

300 lb/ft

A B

3 ft 6 ft1.5 ft

w2

w1

395

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637. Wind has blown sand over a platform such that theintensity of the load can be approximated by the function

Simplify this distributed loading to anequivalent resultant force and specify its magnitude andlocation measured from A.

w = 10.5x32 N>m.

x

w

A

10 m

500 N/m

w (0.5x3) N/m

638. Wet concrete exerts a pressure distribution alongthe wall of the form. Determine the resultant force of thisdistribution and specify the height h where the bracing strutshould be placed so that it lies through the line of action ofthe resultant force. The wall has a width of 5 m.

4 m

h

(4 ) kPap1/2z

8 kPa

z

p

396

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639. Replace the distributed loading with an equivalentresultant force, and specify its location on the beammeasured from point A.

w

xA

B

4 m

8 kN/mw (4 x)212

397

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*640. Replace the loading by an equivalent resultantforce and couple moment at point A.

60

6 ft

50 lb/ft50 lb/ft

100 lb/ft

4 ft

A

B

398

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641. Replace the loading by an equivalent resultantforce and couple moment acting at point B.

60

6 ft

50 lb/ft50 lb/ft

100 lb/ft

4 ft

A

B

399

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642. Determine the moment of inertia of the area aboutthe axis.x

y

x

2 m

2 m

y 0.25 x3

400

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643. Determine the moment of inertia of the area aboutthe axis.y

y

x

2 m

2 m

y 0.25 x3

401

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*644. Determine the moment of inertia of the area aboutthe axis.x

y

x

y2 x3 1 m

1 m

402

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645. Determine the moment of inertia of the area aboutthe axis.y

y

x

y2 x3 1 m

1 m

403

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646. Determine the moment of inertia of the area aboutthe axis.x

y

x

y2 2x

2 m

2 m

404

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647. Determine the moment of inertia of the area aboutthe axis.y

y

x

y2 2x

2 m

2 m

405

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*648. Determine the moment of inertia of the area aboutthe axis.x

y

xO

y 2x4 2 m

1 m

406

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649. Determine the moment of inertia of the area aboutthe axis.y

y

xO

y 2x4 2 m

1 m

407

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650. Determine the polar moment of inertia of the areaabout the axis passing through point .Oz

y

xO

y 2x4 2 m

1 m

408

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651. Determine the moment of inertia of the area aboutthe x axis.

y

x

2 in.

8 in.

y x3

*652. Determine the moment of inertia of the area aboutthe y axis.

y

x

2 in.

8 in.

y x3

409

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

654. Determine the moment of inertia of the area aboutthe y axis.

x

y

1 in.

2 in.

y 2 2 x 3

653. Determine the moment of inertia of the area aboutthe x axis.

x

y

1 in.

2 in.

y 2 2 x 3

410

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655. Determine the moment of inertia of the area aboutthe x axis. Solve the problem in two ways, using rectangulardifferential elements: (a) having a thickness of dx, and (b) having a thickness of dy.

1 in. 1 in.

4 in.

y 4 4x2

x

y

411

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*656. Determine the moment of inertia of the area aboutthe y axis. Solve the problem in two ways, using rectangulardifferential elements: (a) having a thickness of dx, and (b) having a thickness of dy.

1 in. 1 in.

4 in.

y 4 4x2

x

y

412

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

657. Determine the moment of inertia of the triangulararea about the x axis.

y (b x)hb

y

x

b

h

658. Determine the moment of inertia of the triangulararea about the y axis.

y (b x)hb

y

x

b

h

413

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659. Determine the distance to the centroid of thebeams cross-sectional area; then find the moment of inertiaabout the axis.x

y

2 in.

4 in.

1 in.1 in.

Cx

x

y

y

6 in.

414

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

414

*660. Determine the moment of inertia of the beamscross-sectional area about the x axis.

2 in.

4 in.

1 in.1 in.

Cx

x

y

y

6 in.

661. Determine the moment of inertia of the beamscross-sectional area about the y axis.

2 in.

4 in.

1 in.1 in.

Cx

x

y

y

6 in.

2 in.

4 in.

1 in.1 in.

Cx

x

y

y

6 in.

2 in.

4 in.

1 in.1 in.

Cx

x

y

y

6 in.

415

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

662. Determine the moment of inertia of the beamscross-sectional area about the axis.x

y

x

15 mm15 mm60 mm60 mm

100 mm

100 mm

50 mm

50 mm

15 mm

15 mm

416

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663. Determine the moment of inertia of the beamscross-sectional area about the axis.y

y

x

15 mm15 mm60 mm60 mm

100 mm

100 mm

50 mm

50 mm

15 mm

15 mm

417

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y

x

150 mm

300 mm

150 mm

100 mm

100 mm

75 mm

*664. Determine the moment of inertia of the compositearea about the axis.x

418

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665. Determine the moment of inertia of the compositearea about the axis.y

y

x

150 mm

300 mm

150 mm

100 mm

100 mm

75 mm

419

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

666. Determine the distance to the centroid of thebeams cross-sectional area; then determine the moment ofinertia about the axis.x

y

x

xC

y

50 mm 50 mm75 mm

25 mm

25 mm

75 mm

100 mm

_y

25 mm

25 mm

100 mm

420

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

667. Determine the moment of inertia of the beamscross-sectional area about the y axis.

x

xC

y

50 mm 50 mm75 mm

25 mm

25 mm

75 mm

100 mm

_y

25 mm

25 mm

100 mm

*668. Locate the centroid of the composite area, thendetermine the moment of inertia of this area about thecentroidal axis.x

y y

1 in.1 in.

2 in.

3 in.

5 in.x

xy

3 in.

C

421

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

669. Determine the moment of inertia of the compositearea about the centroidal axis.y

y

1 in.1 in.

2 in.

3 in.

5 in.x

xy

3 in.

C

670. Determine the distance to the centroid of thebeams cross-sectional area; then find the moment of inertiaabout the axis.x

y

300 mm

100 mm

200 mm

50 mm 50 mm

y

C

x

y

x

422

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

671. Determine the moment of inertia of the beamscross-sectional area about the x axis.

300 mm

100 mm

200 mm

50 mm 50 mm

y

C

x

y

x

423

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*672. Determine the moment of inertia of the beamscross-sectional area about the y axis.

300 mm

100 mm

200 mm

50 mm 50 mm

y

C

x

y

x

424

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

673. Determine the moment of inertia of the beamscross-sectional area about the axis.x

y

50 mm 50 mm

15 mm115 mm

115 mm

7.5 mmx

15 mm

425

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

674. Determine the moment of inertia of the beamscross-sectional area about the axis.y

y

50 mm 50 mm

15 mm115 mm

115 mm

7.5 mmx

15 mm

426

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

675. Locate the centroid of the cross-sectional area forthe angle. Then find the moment of inertia about the centroidal axis.

xIxy

6 in.2 in.

6 in.

x 2 in.

C x

yy

x

y

*676. Locate the centroid of the cross-sectional areafor the angle. Then find the moment of inertia about the

centroidal axis.yIy

x

6 in.2 in.

6 in.

x 2 in.

C x

yy

x

y

427

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

677. Locate the centroid ( , ) of the area.yx y

x

3 in.1 in.

3 in.6 in.

678. Locate the centroid of the shaded area.y

x

y

a2

a2

a

aa

428

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*680. Determine the moment of inertia of the area aboutthe x axis.

y

4y 4 x2

1 ft

x2 ft

679. Determine the moment of inertia of the area aboutthe y axis.

y

4y 4 x2

1 ft

x2 ft

429

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

681. If the distribution of the ground reaction on thepipe per foot of length can be approximated as shown,determine the magnitude of the resultant force due to thisloading.

2.5 ft

50 lb/ft

25 lb/ft

w 25 (1 cos u) lb/ft

u

430

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

682. Determine the moment of inertia of the beamscross-sectional area about the x axis which passes throughthe centroid C.

Cx

y

d2

d2

d2

d2 60

60

683. Determine the moment of inertia of the beamscross-sectional area about the y axis which passes throughthe centroid C.

Cx

y

d2

d2

d2

d2 60

60

431

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

*684. Determine the moment of inertia of the area aboutthe x axis. Then, using the parallel-axis theorem, find themoment of inertia about the axis that passes through thecentroid C of the area. .y = 120 mm

x

1200

200 mm

200 mm

y

x

xy

Cy x2

432

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

685. Determine the moment of inertia of the beamscross-sectional area with respect to the axis passingthrough the centroid C.

x

0.5 in.

0.5 in.

4 in.

2.5 in.C x

0.5 in.

_y

686. The distributed load acts on the beam as shown.Determine the magnitude of the equivalent resultant forceand specify where it acts, measured from point A. w (2x2 4x 16) lb/ft

xB

A

w

4 ft

433

2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currentlyexist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.

687. The distributed load acts on the beam as shown.Determine the maximum intensity . What is themagnitude of the equivalent resultant force? Specify whereit acts, measured from point B.

wmaxw (2x2 4x 16) lb/ft

xB

A

w

4 ft

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