Four small spheres, each of which you can regard as a point of mass m = 0.100 kg

Question

Four small spheres, each of which you can regard as a point of mass m = 0.100 kg, are arranged in a square d = 0.200 m on a side and connected by light rods (see figure below).

(a) Find the moment of inertia of the system about an axis through the center of the square, perpendicular to its plane (an axis through point O in the figure).
kg · m2

(b) Find the moment of inertia of the system about an axis bisecting two opposite sides of the square (an axis along the line AB in the figure).
kg · m2

(c) Find the moment of inertia of the system about an axis that passes through the centers of the upper left and lower right spheres and through point O.
kg · m2

Explanation / Answer

here,

m = 0.1 kg

d = 0.2 m

a)

for the axis passing through centre and perpendicular to the plane

the distance of all masses are equal ,x = d/sqrt2 m

moment of inertia , I = 4 * m * x^2

I = 4 * 0.1 * (0.2/sqrt 2)^2

I = 0.008 kg m^2

moment of inertia is 0.008 kg m^2

b)

for the axis bisecting two opposite sides of square

the distance of all masses are equal ,x = d/2 m

moment of inertia , I = 4 * m * x^2

I = 4 * 0.1 * (0.2/ 2)^2

I = 0.004 kg m^2

moment of inertia is 0.004 kg m^2

c)

for the axis that passes through the centers of the upper left and lower right spheres and through point O.

the distance of other two masses are equal ,x = d/sqrt2 m

the distance of upper left and lower right spheres = 0 m

moment of inertia , I = 2 * m * x^2

I = 2 * 0.1 * (0.2/sqrt 2)^2

I = 0.004 kg m^2

moment of inertia is 0.004 kg m^2

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