Explanation Given information: The radius ( r ) of the quarter circle plate is 300 mm. Assuming the acceleration due to gravity ( g ) is 9.81 m/s 2 Calculation: Show the centroid of the quadratic circle as in Figure (1). Write the centroid formula for both x and y axis the quadratic circle: x ¯ = 4 r 3 π y ¯ = 4 r 3 π Show the distance relationship of the thin uniform plate as in Figure (2). Refer Figure (1), Find the distance between PG : P G = x ¯ 2 + y ¯ 2 Substitute 4 r 3 π for x ¯ and 4 r 3 π for y ¯ . P G = ( 4 r 3 π ) 2 + ( 4 r 3 π ) 2 = 2 ( 4 r 3 π ) 2 = 4 r 2 3 π Thus, the distance between PX is 4 r 3 π . Find the distance between OX : O X = r − 4 r 3 π = r ( 1 − 4 3 π ) Convert radius ( r ) from millimeter to meter. r = 300 mm × 1 m 1000 mm = 0.3 m Calculate the distance ( d ) by applying Pythagoras theorem for the triangle OXG . d = y ¯ 2 + O X 2 Substitute r ( 1 − 4 3 π ) for OX and 4 r 3 π for y ¯ . d = ( 4 r 3 π ) 2 + ( r ( 1 − 4 3 π ) ) 2 = ( 4 r 3 π ) 2 + r 2 ( 1 − 4 3 π ) 2 Substitute 0.3 m for r . d = ( 4 ( 0.3 ) 3 π ) 2 + ( 0.3 ) 2 ( 1 − 4 3 π ) 2 = 0.01621 + 0.029817 = 0.21454 m Write the expression for the mass moment of inertia ( I 0 ) : I 0 = [ m r 2 2 − m ( 4 r 2 3 π ) 2 ] + m d 2 Substitute 0

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

12th Edition

ISBN: 9781260916942

Author: BEER

Publisher: MCG

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Free undamped vibrations

Whenever an object/system displaces in either side of its equilibrium position with the help of an external/internal means, then the object/system moves regularly or randomly on both sides of its equilibrium position. This motion is called vibration. Gen…

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Chapter 19.3, Problem 19.76P

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The period (τn) of the small oscillation of the plate.

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Chapter 19.3, Problem 19.75P

Chapter 19.3, Problem 19.77P

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The period ( τ n ) of the small oscillation of the plate.

Solution Summary: The author calculates the distance relationship of the thin uniform plate as shown in Figure (1).

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The period (τn) of the small oscillation of the plate.

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Chapter 19 Solutions