Question

6. Consider a two dimensional mass distribution in the shape of a thin pizza slice: The mass is uniformly distributed and the pizza slice has a radius R and total angle o. Basically you can think of this as a slice of a disk of radius R and angle φ。. a) What is the mass per unit area σ in terms of M, R, and φ? b) Calculate the center of mass of the pizza slice. Check that your answer makes sense if we set a equal to 2π. c) Where is the center of mass for a half-disk? That is, where is the center of mass if φο-π? In particular, is yCM greater than, equal to, or less than "? Does your answer make sense? d) Suppose the pizza slice is rotated in the plane of the paper (the z -y plane) around the origin, what is the moment of inertia of the pizza slice? e) How does your answer for part d) compare to a complete disk with the same total mass M? How does it compare to a complete disk with the same mass per unit area? Note: Remember that in polar coordinates x-r cos, y-r sino, and dA-rdodr where r is the radial distance from the origin and φ is the angle up from the +x-aris.

Answer 1

mass he (x,y) Thn b) ld centu 0

3n 刀一中 2 2 200 MJ