7 STATICS ENGINEERS MECHANICS Theorems of Pappus-Guldinus CHAPTER Lecture Notes: Professor A. Salam Al-Ammri Suhad Ibraheem Mohammed Al-Khwarizmi College of Engineering University of Baghdad asalaam.kecbu@uobaghdad.edu.iq Theorems of Pappus-Guldinus 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Contents Introduction Theorms of Pappus-Guldinus Sample Problem 7.3.1 Sample Problem 7.3.2 Sample Problem 7.3.3 Sample Problem 7.3.4 Sample Problem 7.3.5 Sample Problem 7.3.6 Sample Problem 7.3.7 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Introduction Theorems of Pappus-Guldinus are useful to find a surface area or volume of revolution. Surface area (As) of revolution Volume (V) of revolution 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Theorems of Pappus-Guldinus Surface of revolution is generated by rotating a plane curve about a fixed axis. Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Theorems of Pappus-Guldinus Body of revolution is generated by rotating a plane area about a fixed axis. Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.1 The circular arc is rotated through 360o about the y-axis.Determine the outer surface area of the resulting body, which is a portion of a sphere. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.2 The body shown in cross section is a complete circular ring formed by revolving the octagonal area about the z-axis. The entire surface is to be covered with a special coating . Determine this surface area. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.3 The two circular arcs AB and BC are revolved about the vertical axis to obtain the surface of revolution shown. Compute the area A of the outside of this surface. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.4 The outside diameter of a pulley is 0.8 m, and the cross section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel is ρ=7.85x103 kg/m3,determine the mass and weight of the rim. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.5 Calculate the mass m of concrete required to constructthe arched dam shown. Concrete has a density of 2.40 Mg/m3. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.6 A hand-operated control wheel made of aluminum has the proportions shown in the cross-sectional view. The area of the total section shown is 15,200 mm2, and the wheel has a mass of 10 kg. Calculate the distance to the centroid of the half-section. The aluminum has a density of 2.69 Mg/m3. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad Sample Problem 7.3.7 A steel die, shown in section ,has the form of a solid generated by revolving the shaded area around the z-axis.Calculate the mass of the die. 2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad