Moment of Inertia of a Dumbbell

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Rotational Motion Beginner rotational motion

Source: High school physics (Chinese)

Problem Sets:

Rigid body - 1111

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.

  1. 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?
  2. 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:

$$I_2 = m(0)^2 + m(L)^2 = mL^2$$ $$I_2 = (0.2 \text{ kg})(0.8 \text{ m})^2 = 0.128 \text{ kg m}^2$$