Question

an object consistes of balls and rigid (massless) rods as shown above. the distance between axis 1 and axis 2 is 0.3 m. all three of the balls have a mass of 0.5 kg. what is the moment of inertia about axis

Answer 1

The moment of inertia about axis 1 is 0.05625 kgm².

How to find moment of inertia?

To calculate the moment of inertia of an object consisting of point masses connected by massless rods about a given axis, use the following formula:

`\[ I = \sum m_i r_i^2 \]`

Where

I = moment of inertia,

`\( m_i \)` = mass of the i-th point mass, and

`\( r_i \)` = perpendicular distance from the i-th point mass to the axis of rotation.

Assume that axis 1 and axis 2 are parallel, and the three balls are aligned perpendicular to these axes. If the rods are massless, they do not contribute to the moment of inertia.

Also assume that one ball is on axis 1, the second ball is on axis 2, and the third ball is halfway between them (0.15 m from each axis):

The ball on axis 1 contributes 0 because it is on the axis of rotation.

The ball on axis 2 is 0.3 m from axis 1.

The ball in the middle is 0.15 m from axis 1.

Now calculate the moment of inertia for each ball about axis 1:

`\[ I = m \cdot r_1^2 + m \cdot r_2^2 + m \cdot r_3^2 \]`

Where

m = mass of each ball (0.5 kg),

r₁ = distance of the first ball from axis 1 (0 m),

r₂ = distance of the second ball from axis 1 (0.3 m), and

r₃ = distance of the third ball from axis 1 (0.15 m).

Plugging in these values:

`\[ I = 0.5 \cdot 0^2 + 0.5 \cdot 0.3^2 + 0.5 \cdot 0.15^2 \]`

`\[ I = 0.5 \cdot 0 + 0.5 \cdot 0.09 + 0.5 \cdot 0.0225 \]`

`\[ I = 0 + 0.045 + 0.01125 \]`

`\[ I = 0.05625 \text{ kgm}^2 \]`

Therefore the moment of inertia about axis 1 with the given assumptions is 0.05625 kgm².