Problem 9-76

Determine the location xc of the centroid of the solid made from a hemisphere, cylinder, and cone.

Given:

a 80mm:=b 60mm:=c 30mm:=d 30mm:=

Solution:

V13 d2 a d2 b+ 2

3 d3+:=

xc1V

13 d2 a 3a

4 d2 b a b

2+

+23 d3 a b+ 3c

8+

+

:=

xc 105.2 mm=

Problem 9-77

The buoy is made from two homogeneous cones each having radius r. Find the distance zc to thebuoy's center of gravity G.

Given:

r 1.5m:=h 1.2m:=a 4m:=

Solution:

zc

3

r2 a a4

3

r2 h h4

3

r2 a h+( ):= zc 0.700 m=

Problem 9-78

The buoy is made from two homogeneous cones each having radius r. If it is required that thebuoy's center of gravity G be located at zc,determine the height h of the top cone.

Given:

zc 0.5m:=r 1.5m:=a 4m:=

Solution:

Guess h 1ft:=

Given zc

3

r2 a a4

3

r2 h h4

3

r2 a h+( )= h Find h( ):= h 2.00 m=

Problem 9-79

Locate the center of mass zc of the forked lever, which is made from a homogeneous material andhas the dimensions shown.

Given:

a 12.5mm:=b 62.5mm:=c 50mm:=d 75mm:=e 12.5mm:=

Solution:

V b a2 2 e a d+ 2

c e+( )2 c2 a+:=

zc1V

b a2 b2

2 e a d b e+ c+ d2

++

a2

c e+( )2 b c+ e+ 4 c e+3

+

a2

c2 b c+ e+ 4 c3

+

...

:=

zc 108 mm=

Problem 9-80

A triangular plate made of homogeneous material has a constant thickness which is very small. Ifit is folded over as shown, determine the location yc of the plate's center of gravity G.

Given

a 6cm:=

b 3cm:=

c 1cm:=

d 3cm:=

e 1cm:=

f 3cm:=

Solution:

yc

2 d b b2

12

2 c b 2 b3

+ 12

2 e f f3

+

2 d b 12

2 c b+ 12

2 d a f+( )+:= yc 0.75 cm=

Problem 9-81

A triangular plate made of homogeneous material has a constant thickness which is very small.If it is folded over as shown, determine the location zc of the plate's center of gravity G.

Given

a 6cm:=

b 3cm:=

c 1cm:=

d 3cm:=

e 1cm:=

f 3cm:=

Solution:

zc

12

2 e f a 2 e a a2

+ 12

2 d e( ) a a3

+

2 d b 12

2 c b+ 12

2 d a f+( )+:= zc 1.625 cm=

Problem 9-82

Each of the three homogeneous plates welded to the rod has a density and a thickness a.Determine the length l of plate C and the angle of placement, , so that the center of mass of theassembly lies on the y axis. Plates A and B lie in the xy and zy planes, respectively.

Units Used:

Mg 1000kg:=Given:

a 10mm:= f 100mm:=b 200mm:= g 150mm:=c 250mm:= e 150mm:=

6 Mgm3

:=

Solution: The thickness and density are uniform

Guesses 10deg:= l 10mm:=Given

b f f2

g l g2

cos ( ) 0= c e e2

g l g2

sin ( )+ 0=

l

Find l ,( ):= l 265 mm= 70.4 deg=

Problem 9-83

The assembly consists of a wooden dowel rod of length L and a tight-fitting steel collar.Determine the distance xc to its center of gravity if the specific weights of the materials are wand st.The radii of the dowel and collar are shown.Given:

L 20cm:=w 24

kN

m3:=

st 78kN

m3:=

a 5cm:=b 5cm:=r1 1cm:=r2 2cm:=

Solution:

xc

w r12 LL2

st r22 r12 b a b2+

+w r12 L st r22 r12 b+

:= xc 8.23 cm=

Problem 9-84

Determine the surface area and the volume of the ring formed by rotating the square about thevertical axis.

Given:

45deg:=

Solution:

A 2 2 b a2

sin ( ) a

+

2 2 b a2

sin ( )+ a

+

...=

A 8ba=

V 2ba2=

Problem 9-85

The anchor ring is made of steel having specific weight st. Determine the surface area of thering. The cross section is circular as shown.

Given

st 78kN

m3:=

a 4cm:=

b 8cm:=

Solution:

A 2 a2

b a4

+ 2

b a4

:= A 118.4 cm2=

Problem 9-86

Using integration, determine both the area and the distance yc to the centroid of the shadedarea.Then using the second theorem of PappusGuldinus, determine the volume of the solidgenerated by revolving the shaded area about the x axis.

Given

a 1 m:=b 2 m:=c 2 m:=

Solution:

A

0

c

yayc

2b+

d:= A 3.33 m2=

yc1A

0

c

yy ayc

2b+

d:= yc 1.20 m=

V 2 yc A:= V 25.1 m3=

Problem 9-87

The grain bin of the type shown is manufactured by Grain Systems, Inc. Determine the requiredsquare footage of the sheet metal needed to form it, and also the maximum storage capacity(volume) within it.

Given:

a 9m:=b 6m:=c 13.5m:=

Solution:

A 2 a c 2 a2

a2 b2++:=

A 1069 m2=

V 2 a c a2

2 a3

12

a b+:=

V 3944 m3=

Problem 9-88

Determine the surface area and the volume of the conical solid.

Solution:

A 2 a 32

a2

2 =

A 3 a2=

V 212

a2

32

a 36

a 2 =

V4

a3=

Problem 9-89

Sand is piled between two walls as shown. Assume the pile to be a quarter section of a cone andthat ratio p of this volume is voids (air space). Use the second theorem of Pappus-Guldinus todetermine the volume of sand.

Given:

r 3m:=h 2m:=p 0.26:=

Solution:

V 1 p( ) 2

r3

h r2

:=

V 3.49 m3=

Problem 9-90

The rim of a flywheel has the cross section A-A shown. Determine the volume of materialneeded for its construction.

Given:

r 300mm:=a 20mm:=b 40mm:=c 20mm:=d 60mm:=

Solution:

V 2 r b+ c2

+ d c 2 r

b2

+ b a+:=

V 4.25 106 mm3=

Problem 9-91

The Gates Manufacturing Co. produces pulley wheels such as the one shown. Determine theweight of the wheel if it is made from steel having a specific weight .Given:

a 25mm:=c 12.5mm:=d 25mm:=e 25mm:=f 6.25mm:=

b 2 c d+ e+( ):=

78 kNm3

:=

Solution:

W 2 d a c d2

+ c d+

a3

+

a f2

e+

:= W 12.92 N=

Problem 9-92

The Gates Manufacturing Co. produces pulley wheels such as the one shown. Determine the totalsurface area of the wheel in order to estimate the amount of paint needed to protect its surfacefrom rust.

Given:

a 25mm:=c 12.5mm:=d 25mm:=e 25mm:=f 6.25mm:=

b 2 c d+ e+( ):=

Solution:

A 2 f c d+( ) a c+ 2 d e+( ) c d e+2

++ 2 e

2 a f2

2+ c d+ e

2+

+

:=

A 43774 mm2=

Problem 9-93

Determine the volume of material needed to make the casting.

Given:

r1 4cm:=

r2 6cm:=

r3 r2 r1:=

Solution:

V 2 2 4

r2

24 r23

2 r2 2 r3( )

r22

+ 2

2

r3

2 r24 r33

:=

V 1403 cm3=

Problem 9-94

A circular sea wall is made of concrete. Determine the total weight of the wall if the concretehas a specific weight c.

Given:

c 24kN

m3:=

a 18m:=b 4.5m:=c 2.4m:=d 9m:= 50deg:=

Solution:

W c a23

b c( )+

12

d b c( ) a b+ c2

d c+

:=

W 13476 kN=

Problem 9-95

Determine the surface area of the tank, which consists of a cylinder and hemispherical cap.

Given:

a 4m:=b 8m:=

Solution:

A 2 a b 2 a a

2+

:=

A 302 m2=

Problem 9-96

Determine the volume of the tank, which consists of a cylinder and hemispherical cap.

Given:

a 4m:=b 8m:=

Solution:

V 2 4 a3

a24

a2

b a+

:=

V 536 m3=

Problem 9-97

Determine the surface area of the silo which consists of a cylinder and hemispherical cap.Neglect the thickness of the plates.

Given:

a 3m:=b 3m:=c 24m:=

Solution:

A 2 2aa2

a c+:=

A 509 m2=

Problem 9-98

Determine the volume of the silo which consists of a cylinder and hemispherical cap. Neglect thethickness of the plates.

Given :

a 10m:=b 3m:=c 24m:=

Solution:

V 2 4 a3

a24

c a a2

+

:=

V 9634 m3=

Problem 9-99

The process tank is used to store liquids during manufacturing. Estimate both the volume of thetank and its surface area. The tank has a flat top and the plates from which the tank is madehave negligible thickness.

Given:

a 4m:=b 6m:=c 3m:=

Solution:

V 2 c3

c a2

c2

c b+:=

V 207 m3=

A 2 c2

c c b+ c2

a2 c2++:=

A 188 m2=

Problem 9-100

Determine the height h to which liquid should be poured into the cup so that it contacts half thesurface area on the inside of the cup. Neglect the cup's thickness for the calculation.

Given :

a 30mm:=b 50mm:=c 10mm:=

Solution :

Total area

Atotal 2 cc2

a c+2

b2 a c( )2++:=

Guess h 1mm:= e 1mm:=Given

a cb

e ch

=

Atotal2

2 c c2

e c+2

h2 e c( )2++=

e

h

Find e h,( ):= e 21.94 mm= h 29.9 mm=

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