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Problem 3 (a) Consider a homogeneous cylinder of mass M, radius R and height h. Show that the inertia tensor in the principal axis system has the form (M(R² + h¹²) 0 0 I = 0 0 M (R² + h¹²) 0 0 R2 (b) Consider a right circular solid cone with radius r, height h and mass m. The density of the cone is constant: Calculate the centre of mass in the Cartesian coordinate system in which the base of the cone lies on the (x, y) plane centered at the origin (see figure). (c) Calculate the moment of inertia of the cone about its symmetry axis. (d) Find the principal axes of the cone. Is there any ambiguity in choosing the principal axis system?

Problem 3 (a) Consider a homogeneous cylinder of mass M, radius R and height h. Show that the inertia tensor in the principal axis system has the form (M(R² + h¹²) 0 0 I = 0 0 M (R² + h¹²) 0 0 R2 (b) Consider a right circular solid cone with radius r, height h and mass m. The density of the cone is constant: Calculate the centre of mass in the Cartesian coordinate system in which the base of the cone lies on the (x, y) plane centered at the origin (see figure). (c) Calculate the moment of inertia of the cone about its symmetry axis. (d) Find the principal axes of the cone. Is there any ambiguity in choosing the principal axis system?

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Problem 3 (a) Consider a homogeneous cylinder of mass M, radius R and height h. Show that the inertia tensor in the principal axis system has the form 4 (R² + ²) 0 0 Jab = 0 0 (R² +²) 0 0 MR² (b) Consider a right circular solid cone with radius r, height h and mass m. The density of the cone is constant: Z Calculate the centre of mass in the Cartesian coordinate system in which the base of the cone lies on the (x, y) plane centered at the origin (see figure). (c) Calculate the moment of inertia of the cone about its symmetry axis. (d) Find the principal axes of the cone. Is there any ambiguity in choosing the principal axis system?
Transcribed Image Text:Problem 3 (a) Consider a homogeneous cylinder of mass M, radius R and height h. Show that the inertia tensor in the principal axis system has the form 4 (R² + ²) 0 0 Jab = 0 0 (R² +²) 0 0 MR² (b) Consider a right circular solid cone with radius r, height h and mass m. The density of the cone is constant: Z Calculate the centre of mass in the Cartesian coordinate system in which the base of the cone lies on the (x, y) plane centered at the origin (see figure). (c) Calculate the moment of inertia of the cone about its symmetry axis. (d) Find the principal axes of the cone. Is there any ambiguity in choosing the principal axis system?
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