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 9.2, Problem 9.48P

9.47 and 9.48 Determine the polar moment of inertia of the area shown with respect to (a) point O, (b) the centroid of the area.

Chapter 9.2, Problem 9.48P, 9.47 and 9.48 Determine the polar moment of inertia of the area shown with respect to (a) point O,

Fig. P9.48

(a)

Expert Solution
Check Mark
To determine

Find the polar moment of inertia of the area with respect to point O.

Answer to Problem 9.48P

The polar moment of inertia of the area with respect to point O is 12.16×106mm4_.

Explanation of Solution

Calculation:

Sketch the cross section as shown in Figure 1.

VECTOR MECHANICS FOR ENGINEERS W/CON >B, Chapter 9.2, Problem 9.48P

Refer to Figure1.

It is divided into 4 parts as shown above.

Find the area of section 1 ellipsoid using the relation:

A1=π2(ar) (1)

Substitute 84mm for a and 42mm for r in Equation (1).

A1=π2(84)(42)=5,541.76mm2

Find the area of section 2 ellipsoid using the relation:

A2=π2(ro)2 (2)

Here, ro is radius of outer section.

Substitute 42mm for r in Equation (2).

A2=π2(42)2=2,770.88mm2

Find the area of section 3 ellipsoid using the relation:

A3=π2(ar) (3)

Substitute 54mm for a and 27mm for r in Equation (3).

A3=π2(54)(27)=2,290.22mm2

Find the area of section 4 ellipsoid using the relation:

A4=π2(ri)2 (4)

Substitute 27mm for r in Equation (4).

A4=π2(27)2=1,145.11mm2

Find the total are of section (A) as shown below:

A=A1+A2+A3+A4 (5)

Substitute 5,541.76mm2 for A1, 2,770.88mm2 for A2, 2,290.22mm2 for A3, and 1,145.11mm2 for A4 in Equation (5).

A=5,541.76+2,770.882,290.221,145.11=4,877.32mm2

Find the centroid (x1) section 1 as shown below:

x1=112π=35.6507mm

Find the centroid (x2) section 2 as shown below:

x2=56π=17.8254mm

Find the centroid (x3) section 3 as shown below:

x3=72π=22.9183mm

Find the centroid (x4) section 4 as shown below:

x4=36π=11.4592mm

Find the centroid (x¯) using the relation as follows:

x¯=A1x1+A2x2+A3x3+A4x4A1+A2+A3+A4 (6)

Substitute 5,541.76mm2 for A1, 2,770.88mm2 for A2, 2,290.22mm2 for A3, 1,145.11mm2 for A4, 35.6507mm for x1, and 17.8254mm for x2, 22.9183mm for  x3, 11.4592mm for x4 , in Equation (6).

x¯=5,541.76(35.6507)+2,770.88(17.8254)2,290.22(22.9183)1,145.11(11.4592)5,541.76+2,770.882,290.221,145.11=197,567.62+49,392.044+52,487.94913,122.04,877.32=22.3094mm

Find the polar moment of inertia (JO)1 section 1 as shown below:

(JO)1=π8(aro)[a2+ro2] (7)

Substitute 84mm for a and 42mm for ro in Equation (7).

(JO)1=π8(84×42)[842+422]=π8(84×42)8,820=12,219,601.62=12.21960×106mm4

Find the polar moment of inertia (JO)2 section 2 as shown below:

(JO)2=π4ro2 (8)

Substitute 42mm for ro in Equation (8).

(JO)2=π4(42)2=2.444392×106mm4

Find the polar moment of inertia (JO)3 section 3 as shown below:

(JO)3=π8(ari)[a2+ri2] (9)

Substitute 54mm for a and 27mm for ri in Equation (9).

(JO)3=π8(54×27)[542+272]=572.555[3,645]=2.08696×106mm4

Find the polar moment of inertia (JO)2 section 2 as shown below:

(JO)4=π4ri2 (10)

Substitute 27mm for ri in Equation (10).

(JO)4=π4(27)2=0.41739×106mm4

Find the total moment of inertia (JO) using the relation:

(JO)=(JO)1+(JO)2(JO)3(JO)4 (11)

Substitute 12.21960×106mm4 for (JO)1, 2.444392×106mm4 for (JO)2, 2.08696×106mm4 for (JO)3, and 0.41739×106mm4 for (JO)4 in Equation (11).

(JO)=12.21960×106+2.444392×1062.08696×1060.41739×106=12,159,652=12.16×106mm4

Thus, the polar moment of inertia of the area with respect to point O is 12.16×106mm4_.

(b)

Expert Solution
Check Mark
To determine

Find the centroid of area.

Answer to Problem 9.48P

The centroid of area is 9.73×106mm4_.

Explanation of Solution

Calculation:

Find the centroid of area using the relation:

JO=JC+AX¯2 (12)

Substitute 12.16×106mm4 for JO, 4,877.32mm2 for A, and 22.3094mm for x¯,and 0.7468in for y¯ in Equation (12).

12.16×106=JC+(4,877.32)(22.3094)212.16×106=JC+2,427,487.66JC=9.73×106mm4

Thus, the centroid of area is 9.73×106mm4_.

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Chapter 9 Solutions

VECTOR MECHANICS FOR ENGINEERS W/CON >B

Ch. 9.1 - Prob. 9.11PCh. 9.1 - Prob. 9.12PCh. 9.1 - 9.12 through 9.14 Determine by direct integration...Ch. 9.1 - 9.12 through 9.14 Determine by direct integration...Ch. 9.1 - Prob. 9.15PCh. 9.1 - Prob. 9.16PCh. 9.1 - Prob. 9.17PCh. 9.1 - Prob. 9.18PCh. 9.1 - Determine the moment of inertia and the radius of...Ch. 9.1 - Prob. 9.20PCh. 9.1 - Determine the polar moment of inertia and the...Ch. 9.1 - Prob. 9.22PCh. 9.1 - Prob. 9.23PCh. 9.1 - 9.23 and 9.24 Determine the polar moment of...Ch. 9.1 - Prob. 9.25PCh. 9.1 - Prob. 9.26PCh. 9.1 - Prob. 9.27PCh. 9.1 - Prob. 9.28PCh. 9.1 - Prob. 9.29PCh. 9.1 - Prove that the centroidal polar moment of inertia...Ch. 9.2 - 9.31 and 9.32 Determine the moment of inertia and...Ch. 9.2 - 9.31 and 9.32 Determine the moment of inertia and...Ch. 9.2 - 9.33 and 9.34 Determine the moment of inertia and...Ch. 9.2 - 9.33 and 9.34 Determine the moment of inertia and...Ch. 9.2 - Determine the moments of inertia of the shaded...Ch. 9.2 - Determine the moments of inertia of the shaded...Ch. 9.2 - Prob. 9.37PCh. 9.2 - Prob. 9.38PCh. 9.2 - Prob. 9.39PCh. 9.2 - Prob. 9.40PCh. 9.2 - 9.41 through 9.44 Determine the moments of inertia...Ch. 9.2 - 9.41 through 9.44 Determine the moments of inertia...Ch. 9.2 - Prob. 9.43PCh. 9.2 - Prob. 9.44PCh. 9.2 - 9.45 and 9.46 Determine the polar moment of...Ch. 9.2 - Prob. 9.46PCh. 9.2 - 9.47 and 9.48 Determine the polar moment of...Ch. 9.2 - 9.47 and 9.48 Determine the polar moment of...Ch. 9.2 - To form a reinforced box section, two rolled W...Ch. 9.2 - Two channels are welded to a d 12-in. steel plate...Ch. 9.2 - Prob. 9.51PCh. 9.2 - Two 20-mm steel plates are welded to a rolled S...Ch. 9.2 - A channel and a plate are welded together as shown...Ch. 9.2 - Prob. 9.54PCh. 9.2 - Two L76 76 6.4-mm angles are welded to a C250 ...Ch. 9.2 - Prob. 9.56PCh. 9.2 - Prob. 9.57PCh. 9.2 - 9.57 and 9.58 The panel shown forms the end of a...Ch. 9.2 - Prob. 9.59PCh. 9.2 - Prob. 9.60PCh. 9.2 - Prob. 9.61PCh. 9.2 - Prob. 9.62PCh. 9.2 - Prob. 9.63PCh. 9.2 - Prob. 9.64PCh. 9.2 - Prob. 9.65PCh. 9.2 - Prob. 9.66PCh. 9.3 - 9.67 through 9.70 Determine by direct integration...Ch. 9.3 - 9.67 through 9.70 Determine by direct integration...Ch. 9.3 - 9.67 through 9.70 Determine by direct integration...Ch. 9.3 - Prob. 9.70PCh. 9.3 - Prob. 9.71PCh. 9.3 - Prob. 9.72PCh. 9.3 - Prob. 9.73PCh. 9.3 - 9.71 through 9.74 Using the parallel-axis theorem,...Ch. 9.3 - Prob. 9.75PCh. 9.3 - 9.75 through 9.78 Using the parallel-axis theorem,...Ch. 9.3 - Prob. 9.77PCh. 9.3 - Prob. 9.78PCh. 9.3 - Determine for the quarter ellipse of Prob. 9.67...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - 9.75 through 9.78 Using the parallel-axis theorem,...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - Determine the moments of inertia and the product...Ch. 9.3 - Prob. 9.85PCh. 9.3 - 9.86 through 9.88 For the area indicated,...Ch. 9.3 - Prob. 9.87PCh. 9.3 - Prob. 9.88PCh. 9.3 - Prob. 9.89PCh. 9.3 - 9.89 and 9.90 For the angle cross section...Ch. 9.4 - Using Mohrs circle, determine for the quarter...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Prob. 9.93PCh. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - Using Mohrs circle, determine the moments of...Ch. 9.4 - For the quarter ellipse of Prob. 9.67, use Mohrs...Ch. 9.4 - 9.98 though 9.102 Using Mohrs circle, determine...Ch. 9.4 - Prob. 9.99PCh. 9.4 - 9.98 though 9.102 Using Mohrs circle, determine...Ch. 9.4 - Prob. 9.101PCh. 9.4 - Prob. 9.102PCh. 9.4 - Prob. 9.103PCh. 9.4 - 9.104 and 9.105 Using Mohrs circle, determine the...Ch. 9.4 - 9.104 and 9.105 Using Mohrs circle, determine the...Ch. 9.4 - For a given area, the moments of inertia with...Ch. 9.4 - it is known that for a given area Iy = 48 106 mm4...Ch. 9.4 - Prob. 9.108PCh. 9.4 - Prob. 9.109PCh. 9.4 - Prob. 9.110PCh. 9.5 - A thin plate with a mass m is cut in the shape of...Ch. 9.5 - A ring with a mass m is cut from a thin uniform...Ch. 9.5 - A thin elliptical plate has a mass m. Determine...Ch. 9.5 - The parabolic spandrel shown was cut from a thin,...Ch. 9.5 - Prob. 9.115PCh. 9.5 - Fig. P9.115 and P9.116 9.116 A piece of thin,...Ch. 9.5 - A thin plate of mass m is cut in the shape of an...Ch. 9.5 - Fig. P9.117 and P9.118 9.118 A thin plate of mass...Ch. 9.5 - Determine by direct integration the mass moment of...Ch. 9.5 - The area shown is revolved about the x axis to...Ch. 9.5 - The area shown is revolved about the x axis to...Ch. 9.5 - Determine by direct integration the mass moment of...Ch. 9.5 - Fig. P9.122 and P9.123 9.123 Determine by direct...Ch. 9.5 - Prob. 9.124PCh. 9.5 - Prob. 9.125PCh. 9.5 - Prob. 9.126PCh. 9.5 - Prob. 9.127PCh. 9.5 - Prob. 9.128PCh. 9.5 - Prob. 9.129PCh. 9.5 - Knowing that the thin cylindrical shell shown has...Ch. 9.5 - A circular hole of radius r is to be drilled...Ch. 9.5 - The cups and the arms of an anemometer are...Ch. 9.5 - Prob. 9.133PCh. 9.5 - Determine the mass moment of inertia of the 0.9-lb...Ch. 9.5 - Prob. 9.135PCh. 9.5 - Prob. 9.136PCh. 9.5 - A 2-mm thick piece of sheet steel is cut and bent...Ch. 9.5 - A section of sheet steel 0.03 in. thick is cut and...Ch. 9.5 - A corner reflector for tracking by radar has two...Ch. 9.5 - A farmer constructs a trough by welding a...Ch. 9.5 - The machine element shown is fabricated from...Ch. 9.5 - Determine the mass moments of inertia and the...Ch. 9.5 - Determine the mass moment of inertia of the steel...Ch. 9.5 - Prob. 9.144PCh. 9.5 - Determine the mass moment of inertia of the steel...Ch. 9.5 - Aluminum wire with a weight per unit length of...Ch. 9.5 - The figure shown is formed of 18-in.-diameter...Ch. 9.5 - A homogeneous wire with a mass per unit length of...Ch. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - Determine the mass products of inertia Ixy, Iyz,...Ch. 9.6 - 9.153 through 9.156 A section of sheet steel 2 mm...Ch. 9.6 - Prob. 9.154PCh. 9.6 - Prob. 9.155PCh. 9.6 - 9.153 through 9.156 A section of sheet steel 2 mm...Ch. 9.6 - Prob. 9.157PCh. 9.6 - Prob. 9.158PCh. 9.6 - Prob. 9.159PCh. 9.6 - Prob. 9.160PCh. 9.6 - Prob. 9.161PCh. 9.6 - For the homogeneous tetrahedron of mass m shown,...Ch. 9.6 - Prob. 9.163PCh. 9.6 - Prob. 9.164PCh. 9.6 - Prob. 9.165PCh. 9.6 - Determine the mass moment of inertia of the steel...Ch. 9.6 - Prob. 9.167PCh. 9.6 - Prob. 9.168PCh. 9.6 - Prob. 9.169PCh. 9.6 - 9.170 through 9.172 For the wire figure of the...Ch. 9.6 - Prob. 9.171PCh. 9.6 - Prob. 9.172PCh. 9.6 - Prob. 9.173PCh. 9.6 - Prob. 9.174PCh. 9.6 - Prob. 9.175PCh. 9.6 - Prob. 9.176PCh. 9.6 - Prob. 9.177PCh. 9.6 - Prob. 9.178PCh. 9.6 - Prob. 9.179PCh. 9.6 - Prob. 9.180PCh. 9.6 - Prob. 9.181PCh. 9.6 - Prob. 9.182PCh. 9.6 - Prob. 9.183PCh. 9.6 - Prob. 9.184PCh. 9 - Determine by direct integration the moments of...Ch. 9 - Determine the moment of inertia and the radius of...Ch. 9 - Prob. 9.187RPCh. 9 - Prob. 9.188RPCh. 9 - Prob. 9.189RPCh. 9 - Two L4 4 12-in. angles are welded to a steel...Ch. 9 - Prob. 9.191RPCh. 9 - Prob. 9.192RPCh. 9 - Prob. 9.193RPCh. 9 - Prob. 9.194RPCh. 9 - Prob. 9.195RPCh. 9 - Determine the mass moment of inertia of the steel...
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