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Chapter 17

# Chapter 17 - Engineering Mechanics Dynamics Chapter 17...

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Engineering Mechanics - Dynamics Chapter 17 Problem 17-1 The right circular cone is formed by revolving the shaded area around the x axis. Determine the moment of inertia I x and express the result in terms of the total mass m of the cone. The cone has a constant density U . Solution: m 0 h x U S rx h § ¨ © · ¸ ¹ 2 ´ µ µ d 1 3 h U S r 2 U 3 m h S r 2 I x 3 10 mr 2 I x 3 m h S r 2 0 h x 1 2 S rx h § ¨ © · ¸ ¹ 2 rx h § ¨ © · ¸ ¹ 2 ´ µ µ d Problem 17-2 Determine the moment of inertia of the thin ring about the z axis. The ring has a mass m . Solution: m 0 2 S T U R ´ µ d 2 S U R U m 2 S R I z m 2 S R 0 2 S T R R 2 ´ µ d mR 2 I z mR 2 Problem 17-3 The solid is formed by revolving the shaded area around the y axis. Determine the radius of gyration k y . The specific weight of the material is J . Given: J 380 lb ft 3 a 3 in b 3 in 504

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Engineering Mechanics - Dynamics Chapter 17 Solution: m 0 b y J S a y 3 b 3 § ¨ ¨ © · ¸ ¸ ¹ 2 ´ µ µ µ d m 0.083 slug I y 0 b y J 1 2 S a y 3 b 3 § ¨ ¨ © · ¸ ¸ ¹ 2 a y 3 b 3 § ¨ ¨ © · ¸ ¸ ¹ 2 ´ µ µ µ d I y 0.201 slug in 2 ¡ k I y m k 1.56 in *Problem 17-4 Determine the moment of inertia I x of the sphere and express the result in terms of the total mass m of the sphere.The sphere has a constant density U . Solution: m r 0 r x U S r 2 x 2 0 + , ´ µ d 4 3 r 3 U S U 3 m 4 S r 3 I x 3 m 4 S r 3 r 0 r x S 2 r 2 x 2 0 + , 2 ´ µ µ d I x 2 5 mr 2 Problem 17-5 Determine the radius of gyration k x of the paraboloid. The density of the material is U . Units Used: Mg 10 6 gm Given: h 200 mm r 100 mm U 5 Mg m 3 505
Engineering Mechanics - Dynamics Chapter 17 Solution: M 0 h x U S xr 2 h § ¨ © · ¸ ¹ ´ µ µ d M 15.708 kg I x 0 h x 1 2 U S xr 2 h § ¨ © · ¸ ¹ 2 ´ µ µ µ d I x 0.052 kg m 2 ¡ k x I x M k x 57.7 mm Problem 17-6 Determine the moment of inertia of the semiellipsoid with respect to the x axis and express the result in terms of the mass m of the semiellipsoid. The material has a constant density U . Solution: m 0 a x U S b 2 1 x 2 a 2 0 § ¨ ¨ © · ¸ ¸ ¹ ´ µ µ µ d 2 3 a U S b 2 U 3 m 2 a S b 2 I x 3 m 2 a S b 2 0 a x 1 2 S b 2 1 x 2 a 2 0 § ¨ ¨ © · ¸ ¸ ¹ ª « « ¬ º » » ¼ 2 ´ µ µ µ d I x 2 5 mb 2 Problem 17-7 Determine the radius of gyration k x of the body. The specific weight of the material is J . Given: J 380 lb ft 3 h 8 in r 2 in 506

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Engineering Mechanics - Dynamics Chapter 17 Solution: M 0 h x J S r x h § ¨ © · ¸ ¹ 1 3 ª « « ¬ º » » ¼ 2 ´ µ µ µ µ d M 0.412 slug I x 0 h x 1 2 J S r x h § ¨ © · ¸ ¹ 1 3 ª « « ¬ º » » ¼ 4 ´ µ µ µ µ d I x 0.589 slug in 2 ¡ k x I x M k x 1.20 in *Problem 17-8 Determine the moment of inertia of the ellipsoid with respect to the x axis and express the result in terms of the mass m of the ellipsoid. The material has a constant density U .
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