01. a). Let 𝐺 = (𝑥̅, 𝑦̅, 𝑧̅) be the center of gravity of an object. The coordinates of the 𝐺 is given by, 𝑥̅= ∭v 𝑥𝜌 𝑑𝑉
Posted: Wed Jul 06, 2022 12:05 pm
01. a). Let 𝐺 = (𝑥̅, 𝑦̅, 𝑧̅) be the center of gravity of an object. The coordinates of the 𝐺 is given by,
𝑥̅= ∭v 𝑥𝜌 𝑑𝑉 ÷ ∭v 𝜌 𝑑𝑉 ,
𝑦̅ = ∭v 𝑦𝜌 𝑑𝑉 ÷ ∭v 𝜌 𝑑𝑉 ,
𝑧̅= ∭v 𝑧𝜌 𝑑𝑉 ÷ ∭v 𝜌 𝑑𝑉
Where, 𝜌 be the mass of unit volume.
Find the center of gravity of the region in the first octant bounded by 𝑥 + 𝑦 + 𝑧 = 1.
b). Let 𝜌 𝑑𝑣 be the mass of an element of a solid of volume 𝑉, where 𝜌 is the mass of unit volume. Then the moment of inertia of the solid of volume 𝑉 about the 𝑥-axis is given by,
𝑀.𝐼.𝑥−𝑎𝑥𝑖𝑠 = ∭v 𝜌(𝑦 2 + 𝑧 2 ) 𝑑V .
Find the moment of inertia of the uniform solid in the form of octant of the ellipsoid 𝑥2 + 𝑦 2 + 𝑧 2 = 4 about the 𝑥- axis.
𝑥̅= ∭v 𝑥𝜌 𝑑𝑉 ÷ ∭v 𝜌 𝑑𝑉 ,
𝑦̅ = ∭v 𝑦𝜌 𝑑𝑉 ÷ ∭v 𝜌 𝑑𝑉 ,
𝑧̅= ∭v 𝑧𝜌 𝑑𝑉 ÷ ∭v 𝜌 𝑑𝑉
Where, 𝜌 be the mass of unit volume.
Find the center of gravity of the region in the first octant bounded by 𝑥 + 𝑦 + 𝑧 = 1.
b). Let 𝜌 𝑑𝑣 be the mass of an element of a solid of volume 𝑉, where 𝜌 is the mass of unit volume. Then the moment of inertia of the solid of volume 𝑉 about the 𝑥-axis is given by,
𝑀.𝐼.𝑥−𝑎𝑥𝑖𝑠 = ∭v 𝜌(𝑦 2 + 𝑧 2 ) 𝑑V .
Find the moment of inertia of the uniform solid in the form of octant of the ellipsoid 𝑥2 + 𝑦 2 + 𝑧 2 = 4 about the 𝑥- axis.