Explanation Find the mass of the small gears ( m o ) using the relation. m o = ρ A o t o = ρ ( π r o 2 ) t o Here, the mass density of the shaft is ρ , the cross sectional area of the small gears is A o , the thickness of the small gear is t o , and the radius of the small gears is r o . Substitute r for r o and t for t o . m o = ρ ( π r 2 ) t = ρ π r 2 t Find the mass moment of inertia ( I o ¯ ) of the small gears using the relation. I o ¯ = 1 2 m o r o 2 Substitute ρ π r 2 t for m o and r for r o . I o ¯ = 1 2 ρ π r 2 t r 2 = 1 2 ρ π r 4 t Find the mass of the large gears ( m ) using the relation. m = ρ A t = ρ ( π r 2 ) t Here, the cross sectional area of the large gears is A , the thickness of the large gear is t , and the radius of the large gears is r . Substitute nr for r . m = ρ ( π ( n r ) 2 ) t = ρ π n 2 r 2 t Find the mass moment of inertia ( I ¯ ) of the large gears using the relation. I ¯ = 1 2 m r 2 Substitute ρ π n 2 r 2 t for m and nr for r . I ¯ = 1 2 ρ π n 2 r 2 t ( n r ) 2 = 1 2 ρ π n 4 r 4 t = n 4 I o ¯ Find the angular velocity of the shaft B ( ω B ) using the relation. ω B = ω A n Find the angular velocity of the shaft C ( ω C ) using the relation. ω C = ω A n 2 The number of revolution takes place in the shaft C is θ = 1 rev . Find the work done ( U 1 → 2 ) for the rigid body using the relation; U 1 → 2 = M o θ Here, the magnitude of couple moment applied at shaft C is M o . Substitute 1 rev for θ . U 1 → 2 = M o ( 1 rev × 2 π rad 1 rev ) = 2 π M o The total kinetic energy at rest ( T 1 ) is zero. T 1 = 0 Find the total kinetic energy ( T 2 ) using the relation. T 2 = 1 2 I A ¯ ω A 2 + 1 2 I B ¯ ω B 2 + 1 2 I C ¯ ω C 2 Substitute I o ¯ for I A ¯ , ( I o ¯ + n 4 I o ¯ ) for I B ¯ , ( ω A n ) for ω B , and ( n 4 I o ¯ ) for I C ¯ , and ( ω A n 2 ) for ω C

VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)

12th Edition

ISBN: 9781260916942

Author: BEER

Publisher: MCG

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Chapter 17.1, Problem 17.13P

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Find the angular velocity of shaft A.

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Chapter 17.1, Problem 17.12P

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Chapter 17 Solutions

VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)

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Find the angular velocity of shaft A .

Solution Summary: The author calculates the angular velocity of shaft A and the mass of the small gears using the relation.

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