## Mass Moment of Inertia

For the point mass figure below, what is the mass moment of inertia?

##
__
__**Hint**

**Hint**

For point masses, the mass moment of inertia is:

$$$I=\sum m_ir_i^2$$$

where
$$m$$
is the mass, and
$$r$$
is the perpendicular distance to the rotation axis.

##
__
__**Hint 2**

**Hint 2**

Ignore the 0.4 m distance. That dimension is there to throw you off.

For point masses, the mass moment of inertia is:

$$$I=\sum m_ir_i^2$$$

where
$$m$$
is the mass, and
$$r$$
is the perpendicular distance to the rotation axis.

The figure has four point masses:

$$$I=m_1r_{1}^{2}+m_2r_{2}^{2}+m_3r_{3}^{2}+m_4r_{4}^{2}$$$

Remember,
$$r$$
is the perpendicular distance the mass is from the rotation axis, not parallel distance, meaning the 0.4 m dimension is there for confusion and can be ignored. Starting from the bottom right point mass and moving clockwise:

$$$I=(0.25kg)(1m)^{2}+(0.25kg)(0.75m)^{2}+(0.65kg)(0.5m)^{2}+(0.25kg)(0.5m)^{2}$$$

$$$=(0.25kg)(1m^2)+(0.25kg)(0.5625m^2)+(0.65kg)(0.25m^2)+(0.25kg)(0.25m^2)$$$

$$$=0.25kg\cdot m^2+0.140625kg\cdot m^2+0.1625kg\cdot m^2+0.0625kg\cdot m^2$$$

$$$I=0.616\:kg\cdot m^2$$$

$$$0.616\:kg\cdot m^2$$$