Explanation Refer Fig. P5.130 and Fig. 1. A horizontal slice element is chosen with a thickness dy . Write the equation for area of the base of the pyramid. A base = k b 2 (I) Here, k = k ( N ) and b is the length of the base. Write the equation from the triangle in figure 1. s b = h − y h s = b h ( h − y ) (II) Here, s is the length of the element in the slice, y is the distance from the base of the triangle to the element and h is the height of the section. Write the expression for the volume of the slice using equation (I). d V = A slice d y = k s 2 d y = k b 2 h 2 ( h − y ) 2 d y (III) Write the expression for y ¯ of the volume. y ¯ V = ∫ y ¯ E L d V y ¯ = ∫ y ¯ E L d V V (IV) Here, y ¯ is the y coordinate of the centroid and y ¯ E L is the y coordinate of the centroid of the element. Write the expression for y ¯ of the element. y ¯ E L = y Conclusion: Calculate the volume using equation (III). V = ∫ 0 h d V = ∫ 0 h k b 2 h 2 ( h − y ) 2 d y = k b 2 h 2 [ − 1 3 ( h − y ) 3 ] 0 h = 1 3 k b 2 h (V) Calculate ∫ y ¯ E L d V for the volume obtained by rotating the shaded area

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Chapter 5.4, Problem 5.130P

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To show that for a regular pyramid of height h, the centroid of the volume of the pyramid is located at a distance h/4 above the base.

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Chapter 5.4, Problem 5.129P

Chapter 5.4, Problem 5.131P

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USE MATLAB ONLY Turbomachienery . GIven: vx = 185 m/s, flow angle = 60 degrees, R = 0.5, U = 150 m/s, b2 = -a3, a2 = -b3 Find: velocity triangle , a. magnitude of abs vel leaving rotor (m/s) b. flow absolute angles (a1, a2, a3) 3. flow rel angles (b2, b3) d. specific work done e. use code to draw vel. diagram Use this code for plot % plots Velocity Tri. in Ch4 function plotveltri(al1,al2,al3,b2,b3) S1L = [0 1]; V1x = [0 0]; V1s = [0 1*tand(al3)]; S2L = [2 3]; V2x = [0 0]; V2s = [0 1*tand(al2)]; W2s = [0 1*tand(b2)]; U2x = [3 3]; U2y = [1*tand(b2) 1*tand(al2)]; S3L = [4 5]; V3x = [0 0]; V3r = [0 1*tand(al3)]; W3r = [0 1*tand(b3)]; U3x = [5 5]; U3y = [1*tand(b3) 1*tand(al3)]; plot(S1L,V1x,'k',S1L,V1s,'r',... S2L,V2x,'k',S2L,V2s,'r',S2L,W2s,'b',U2x,U2y,'g',... S3L,V3x,'k',S3L,V3r,'r',S3L,W3r,'b',U3x,U3y,'g',...... 'LineWidth',2,'MarkerSize',10),... axis([-1 6 -4 4]), ... title('Velocity Triangle'), ... xlabel('x'),yl

The wall of a furnace has a thickness of 5 cm and thermal conductivity of 0.7 W/m-°C. The inside surface is heated by convection with a hot gas at 402°C and a heat transfer coefficient of 37 W/m²-°C. The outside surface has an emissivity of 0.8 and is exposed to air at 27°C with a heat transfer coefficient of 20 W/m²-ºC. Assume that the furnace is inside a large room with walls, floor and ceiling at 27°C. Show the thermal circuit and determine the heat flux through the furnace wall. h₁ T₁ k -L T. sur ho E

Turbomachienery . GIven: vx = 185 m/s, flow angle = 60 degrees, R = 0.5, U = 150 m/s, b2 = -a3, a2 = -b3 Find: velocity triangle , a. magnitude of abs vel leaving rotor (m/s) b. flow absolute angles (a1, a2, a3) 3. flow rel angles (b2, b3) d. specific work done e. use code to draw vel. diagram Use this code for plot % plots Velocity Tri. in Ch4 function plotveltri(al1,al2,al3,b2,b3) S1L = [0 1]; V1x = [0 0]; V1s = [0 1*tand(al3)]; S2L = [2 3]; V2x = [0 0]; V2s = [0 1*tand(al2)]; W2s = [0 1*tand(b2)]; U2x = [3 3]; U2y = [1*tand(b2) 1*tand(al2)]; S3L = [4 5]; V3x = [0 0]; V3r = [0 1*tand(al3)]; W3r = [0 1*tand(b3)]; U3x = [5 5]; U3y = [1*tand(b3) 1*tand(al3)]; plot(S1L,V1x,'k',S1L,V1s,'r',... S2L,V2x,'k',S2L,V2s,'r',S2L,W2s,'b',U2x,U2y,'g',... S3L,V3x,'k',S3L,V3r,'r',S3L,W3r,'b',U3x,U3y,'g',...... 'LineWidth',2,'MarkerSize',10),... axis([-1 6 -4 4]), ... title('Velocity Triangle'), ... xlabel('x'),ylabel('y'), grid

Chapter 5 Solutions

VECTOR MECH. FOR EGR: STATS & DYNAM (LL

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To show that for a regular pyramid of height h , the centroid of the volume of the pyramid is located at a distance h / 4 above the base.

Solution Summary: The author illustrates how the centroid of the volume of a regular pyramid is located at h/4 above the base.

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To show that for a regular pyramid of height h, the centroid of the volume of the pyramid is located at a distance h/4 above the base.

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