Questions: In the figure, two particles, each with mass m=0.80 kg, are fastened to each other, and to a rotation axis at 0, by two thin rods, each with length d=5.7 cm and mass M=1.1 kg. The combination rotates around the rotation axis with angular speed ω=0.28 rad / s. Measured about O, what is the combination's (a) rotational inertia and (b) kinetic energy?

Solution Steps

The rotational inertia of a particle of mass \(m\) at a distance \(r\) from the axis of rotation is given by \(I = mr^2\). For the first particle, the distance from the rotation axis is \(d\), so its rotational inertia is \(I_1 = md^2\). For the second particle, the distance from the rotation axis is \(2d\), so its rotational inertia is \(I_2 = m(2d)^2 = 4md^2\).

The rotational inertia of a rod of mass \(M\) and length \(L\) about its center is given by \(\frac{1}{12}ML^2\). Using the parallel axis theorem, the rotational inertia of a rod about an axis a distance \(x\) away from the center of mass is \(I = I_{cm} + Mx^2\).

For the first rod, the distance from the rotation axis to the center of mass is \(d/2\), so its rotational inertia is \(I_3 = \frac{1}{12}Md^2 + M(\frac{d}{2})^2 = \frac{1}{3}Md^2\).

For the second rod, the distance from the rotation axis to the center of mass is \(\frac{3d}{2}\), so its rotational inertia is \(I_4 = \frac{1}{12}Md^2 + M(\frac{3d}{2})^2 = \frac{7}{3}Md^2\).

The total rotational inertia of the combination is the sum of the rotational inertias of the two particles and the two rods:

\(I_{total} = I_1 + I_2 + I_3 + I_4 = md^2 + 4md^2 + \frac{1}{3}Md^2 + \frac{7}{3}Md^2 = 5md^2 + \frac{8}{3}Md^2\)

Substituting the given values: \(m = 0.80 \, kg\), \(M = 1.1 \, kg\), \(d = 0.057 \, m\),

\(I_{total} = 5(0.80)(0.057)^2 + \frac{8}{3}(1.1)(0.057)^2 = 0.013 + 0.0095 = 0.0225 \, kg \cdot m^2\)

The rotational kinetic energy is given by \(K = \frac{1}{2}I\omega^2\).

Substituting the values, \(I = 0.0225 \, kg \cdot m^2\) and \(\omega = 0.28 \, rad/s\):

\(K = \frac{1}{2}(0.0225)(0.28)^2 = 0.000882 \, J\)

Final Answer