Problems

In Problems 1-10, find (a) the mass, (b) the center of mass, (c) the moment of inertia about the origin, of the given plane object.

11             Find the mass of an object in the shape of a unit square whose density is the sum of the four distances from the sides.

12            Find the mass of an object in the shape of a unit square whose density is the product of the distances from the four sides.

13            An object on the triangle 0 ≤ x ≤ 1, 0 ≤ y ≤ x has density equal to the distance from the hypotenuse y = x. Find the amount of work required to stand the object up (a) on one of the short sides, (b) on the hypotenuse.

14            An object in the shape of a unit square has density equal to the distance to the nearest side. Find the mass and the amount of work needed to stand the object up on a side.

15            An object on the plane region - 1≤x≤1, x²≤y≤l has density ρ[x, y) = 1 + x + _N/y. Find the mass and the work needed to stand the object up on the fiat side.

16            An object on the unit square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 has density ρ(x, y) = ax + by + c. Find the mass and center of mass.

17            The moment of an object of density ρ(x, y) in the region D about the vertical line x = a is defined as

Show that

M_y,x=a = M_y - a·m

where M_y is the moment about the y-axis and m is the mass.

18            The moment of inertia of an object in the region D of density ρ(x, y) about the point P(a, b) is defined as

Show that

I_P = I - 2aM_x - 2bM_y + m(a² + b²)

where I is the moment of inertia about the origin, M_x and M_y are the first moments, and m is the mass.