Respuesta :
Answer:
1)[tex]I_{x}=2013.1gcm^{2}[/tex]
2)[tex]I_{y}=744.75gcm^{2}[/tex]
3)[tex]I_{z}=2747.85gcm^{2}[/tex]
Explanation:
Since the given masses are completely in xy plane we have
Moment of inertia about x axis equals
[tex]I_{x}={\sum y_{i}^{2}m_{i}}[/tex]
Applying the values we get
[tex]I_{x}={26\times 3^{2}+23\times 6^2+15\times (-4.5)^{2}+18\times 6^{2}}\\\\\therefore I_{x}=2013.1gcm^{2}[/tex]
Similarly
Moment of inertia about y axis equals
[tex]I_{y}={\sum x_{i}^{2}m_{i}}[/tex]
Applying the values we get
[tex]I_{x}={26\times 3^{2}+23\times (-3)^2+15\times (-4.5)^{2}+18\times 0^{2}}\\\\\therefore I_{y}=744.75gcm^{2}[/tex]
Now by theorem of perpendicular axis of moment of inertia we have
[tex]I_{z}=I_{x}+I_{y}[/tex]
Appying values we get
[tex]I_{z}=2013.1+744.75=2747.85gcm^{2}[/tex]
The rotational inertia of this collection about the x, y, and z axes are 2013.1 g-cm², 744.75 g-cm² and 2747.85 g-cm² respectively.
What is rotational inertia?
The rotational inertia is the product of mass of the body and the square of the radius of the rotation.
it can be given as,
[tex]I=mr^2[/tex]
The masses and coordinates of four particles are as follows:
26 g, x = 3.0 cm, y = 3.0 cm;
8 g, x = 0 cm, y = 6.0 cm;
15 g, x = -4.5 cm, y = -4.5 cm;
23 g, x = -3.0 cm, y = 6.0 cm.
For all the particles, the rotational inertia about x-axis can be given as using the above formula as,
[tex]I_x=\sum m_i(y_i)^2[/tex]
Put the values as,
[tex]I_x=26(3)^2+18(6)^2+15(-4.5)+23(6)\\I_x=2013.1\rm \; g-cm^2[/tex]
The rotational inertia about x-axis can be given as using the above formula as,
[tex]I_y=\sum m_i(x_i)^2[/tex]
Put the values as,
[tex]I_y=26(3)^2+18(0)^2+15(-4.5)+23(-3)\\I_y=744.75\rm \; g-cm^2[/tex]
The rotational inertia about x-axis is the sum of rotational inertia along xy- axis. Thus,
[tex]I_z=2013.1+744.75\\I_z=2747.85\rm \; g-cm^2[/tex]
Hence, the rotational inertia of this collection about the x, y, and z axes are 2013.1 g-cm², 744.75 g-cm² and 2747.85 g-cm² respectively.
Learn more about rotational inertia here;
https://brainly.com/question/13933086
