Explanation Given: The length of the shaded area is a / 2 . The height of the shaded area is a . Show the area of the differential element as in Figure (1). Using Figure 1, Express the parabolic equation. y = a cos π a x Differentiate the above parabolic equation. d y d x = a ( − sin π a x ) ( π a ) = − π sin π a x Compute the length of the differential element using the formula, d L = d x 2 + d y 2 (I) Here, the length of the differential element is dL . Differentiate the Equation (I) with respect to x . d L = 1 + ( d y d x ) 2 d x (II) Here, the centroid of the differential element along the y axis is y ˜ . Compute the centroid of the differential element along x axis using the formula: x ˜ = x Compute the centroid of the differential element along the y axis using the formula: y ˜ = y Determine the location of the centre of gravity of the uniform rod along x axis ( x ¯ ) using the relation, x ¯ = ∫ L x ˜ d L ∫ L d L (III) Determine the location of the centre of gravity of the uniform rod along y axis ( y ¯ ) using the relation, y ¯ = ∫ L y ˜ d L ∫ L d L (IV) Conclusion: Substitute − π sin π a x for d y d x in Equation (II). d L = 1 + ( − π sin π a x ) 2 d x = 1 + π 2 sin 2 π a x d x Apply the limits from 0 to a / 2 , substitute x for x ˜ and 1 + π 2 sin 2 π a x d x for dL in Equation (III). x ¯ = ∫ 0 a / 2 ( x ) ( 1 + π 2 sin 2 π a x ) d x ∫ 0 a / 2 1 + π 2 sin 2 π a x d x (V) Integrate the numerator part. ∫ 0 a / 2 ( x ) ( 1 + π 2 sin 2 π a x ) d x = { 1 π 4 sin 4 ( π a ) ( 2 5 ( 1 + π 2 sin 2 ( π a ) x ) 5 / 2 − 2 3 ( 1 + π 2 sin 2 ( π a ) x ) 3 / 2 ) } 0 a / 2 = 0.3444 a 2 Integrate the denominator part. ∫ 0 a / 2 1 + π 2 sin 2 π a x d x = 1 π 2 sin 2 π a ⋅ ( 1 + π 2 sin 2 ( π a ) x ) 1 2 + 1 1 2 + 1 = [ 2 3 π 2 sin 2 π a ( 1 + π 2 x sin 2 ( π a ) ) 3 / 2 ] 0 a / 2 = 1.1524 a Substitute 1.1524 a for ∫ 0 a / 2 1 + π 2 sin 2 π a x d x and 0.3444 a 2 for ∫ 0 a / 2 ( x ) ( 1 + π 2 sin 2 π a x ) d x in Equation (V)

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

See similar textbooks

Question

Chapter 9.1, Problem 5P

To determine

The centroid co-ordinates (x¯,y¯) of the uniform rod.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.1, Problem 4P

Chapter 9.1, Problem 6P

Students have asked these similar questions

Sketch and Describe a timing diagram for a 2 stroke diesel engine emphasis on the 2 stroke as my last answer explained 4 stroke please include a diagram or sketch.

A 4 ft 200 Ib 1000 Ib.ft C 2 ft 350 Ib - за в 2.5 ft 150 Ib 250 Ib 375 300 Ib Replace the force system acting on the frame. shown in the figure by a resultant force (magnitude and direction), and specify where its line of action intersects member (AB), measured from point (A).

A continuous flow calorimeter was used to obtain the calorific value of a sample of fuel and the following data collected: Mass of fuel: 2.25 kgInlet water temperature: 11 ° COutlet water temperature 60 ° CQuantity of water: 360 Liters Calorimeter efficiency: 85%Calculate the calorific value of the sample ( kJ / kg ). ive submitted this question twice and have gotten two way different answers. looking for some help thanks

Chapter 9 Solutions

Engineering Mechanics: Statics

Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid of the shaded area. Prob....Ch. 9.1 - Locate the center of mass x of the straight rod if...Ch. 9.1 - Locate the centroid of the homogeneous solid...Ch. 9.1 - Locate the centroid z of the homogeneous solid...Ch. 9.1 - Locate the center of mass of the homogeneous rod...Ch. 9.1 - Prob. 2P Ch. 9.1 - Prob. 3P Ch. 9.1 - Prob. 4P

Knowledge Booster

The centroid co-ordinates ( x ¯ , y ¯ ) of the uniform rod.

The centroid co-ordinates (x¯,y¯) of the uniform rod.

Want to see the full answer?

Chapter 9 Solutions