Source: High school physics (Chinese)
Problem Sets:
Problem
A thin rod of length 80 cm has a 200 g small ball attached to each end. The balls can be treated as point masses, and the mass of the rod is negligible.
- What is the moment of inertia of the setup when it rotates about an axis passing through the center of the rod and perpendicular to it?
- What is the moment of inertia of the setup when it rotates about an axis passing through one of the balls and perpendicular to the rod?
[Q1] 0.064 kg m^2 [Q2] 0.128 kg m^2
The moment of inertia for a system of point masses is $I = \sum m_i r_i^2$, where $m_i$ is the mass of the i-th particle and $r_i$ is its distance from the axis of rotation. Let the mass of each ball be $m = 0.2$ kg and the rod length be $L = 0.8$ m.
[Q1] The axis of rotation is at the center of the rod. Each ball is at a distance of $r = L/2$ from the axis. The total moment of inertia $I_1$ is the sum of the moments of inertia of the two balls.
$$I_1 = m\left(\frac{L}{2}\right)^2 + m\left(\frac{L}{2}\right)^2 = 2m\frac{L^2}{4} = \frac{1}{2}mL^2$$ $$I_1 = \frac{1}{2}(0.2 \text{ kg})(0.8 \text{ m})^2 = 0.064 \text{ kg m}^2$$[Q2] The axis of rotation passes through one of the balls. One ball is at a distance $r_1=0$ from the axis, and the other is at a distance $r_2=L$. The total moment of inertia $I_2$ is:
$$I_2 = m(0)^2 + m(L)^2 = mL^2$$ $$I_2 = (0.2 \text{ kg})(0.8 \text{ m})^2 = 0.128 \text{ kg m}^2$$