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 = –

Question

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?

in progress 0
Helga 4 years 2021-08-12T19:40:17+00:00 1 Answers 44 views 0

Answers ( )

  1. giabao
    0
    2021-08-12T19:41:33+00:00
    Reply

    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

Leave an answer

Browse

Giải phương trình 1 ẩn: x + 2 - 2(x + 1) = -x . Hỏi x = ? ( )