The masses and coordinates of four particles are as follows: 67 g, x = 3.0 cm, y = 3.0 cm; 30 g, x = 0, y = 6.0 cm; 41 g, x = - 4.5 cm, y = - 4.5 cm; 53 g, x = - 3.0 cm, y = 6.0 cm. What are the rotational inertias of this collection about the (a) x, (b) y, and (c) z axes?

Given these 4 particles masses and their coordinates

M1 = 67 g, x1 = 3.0 cm, y1 = 3.0 cm;

M2 = 30 g, x2 = 0, y2 = 6.0 cm;

M3 = 41 g, x3 = - 4.5cm, y3 = - 4.5 cm;

M4 = 53 g, x4 = - 3.0cm, y4 = 6.0 cm.

What is Rotational inertia about x, y, z axis?

Rotation inertia is given as,

I = Σ mi•ri²

Therefore for a four particle system,

I = M1•r1² + M2•r2² + M3•r3² + M4•r4²

a. The moment of inertia about x axis is given as

Ix = Σ mi•yi²

Ix=M1•y1²+M2•y2²+M3•y3²+M4•y4²

Ix=67•3² + 30•6² + 41• (-4.5) ² + 53•6²

Ix = 603 + 1080 + 830.25 + 1908

Ix = 4421.25 g•cm²

b. The moment of inertia about y axis is given as

Iy = Σ mi•xi²

Iy=M1•x1²+M2•x2²+M3•x3²+M4•x4²

Iy=67•3² + 30•0² + 41• (-4.5) ² + 53• (-3) ²

Iy = 603 + 0 + 830.25 + 477

Iy = 1910.25 g•cm².

c. The moment of inertia about z can be calculated using the fact that the distance from z axis is

z = √ (x²+y²)

Then, applying this

Iz = Σ mi•zi²

Then, Iz = Σ mi• (√xi²+yi²) ²

Iz = Σ mi• (xi²+yi²)

Separating the summation

Then,

Iz = Σ mi•xi² + Σ mi•yi²

Since,

Σ mi•xi² = Iy = 1910.25 g•cm²

Σ mi•yi² = Ix = 4421.25 g•cm²

Therefore,

Iz = Ix + Iy

Iz = 1901.25 + 4421.25

Iz = 6331.5 g•cm²