Answer:
Explanation:
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
