A rigid body consists of eight particles, each of mass m, held together by light rods. In a certain coordinate frame the particles are at positions

±a(3, 1, −1), ±a(1, −1, 3), ±a(1, 3, −1), ±a(−1, 1, 3).

Show that, when the body rotates about an axis through the origin, if the angular velocity and angular momentum vectors are parallel then their ratio must be 40ma², 64ma² or 72ma².

Question

A rigid body consists of eight particles, each of mass m, held together by light rods. In a certain coordinate frame the particles are at positions

±a(3, 1, −1), ±a(1, −1, 3), ±a(1, 3, −1), ±a(−1, 1, 3).

Show that, when the body rotates about an axis through the origin, if the angular velocity and angular momentum vectors are parallel then their ratio must be 40ma², 64ma² or 72ma².

Accepted Answer

Because the particles are symmetrically placed in pairs with respect to the origin, the inertia tensor, given by

[latex]I_{i j}=\sum_{ ext {particles }} m\left(r^2 \delta_{i j}-x_i x_j ight),[/latex]

will be twice that calulated for the + signs alone. It components are therefore

[latex]\begin{aligned}& I_{11}=2 m a^2(2+10+10+10)=64 m a^2, \& I_{12}= I_{21}=-2 m a^2(3-1+3-1)=-8 m a^2, \& I_{13}= I_{31}=-2 m a^2(-3+3-1-3)=8 m a^2, \& I_{22}=2 m a^2(10+10+2+10)=64 m a^2, \& I_{23}= I_{32}=-2 m a^2(-1-3-3+3)=8 m a^2, \& I_{33}=2 m a^2(10+2+10+2)=48 m a^2 .\end{aligned}[/latex]

The resulting tensor is

[latex]8 m a^2\left(\begin{array}{ccc}8 & -1 & 1 \-1 & 8 & 1 \1 & 1 & 6\end{array} ight)[/latex]

and its principal moments are 8ma²λ, where

[latex]\begin{aligned}0 & =\left|\begin{array}{ccc}8-\lambda & -1 & 1 \-1 & 8-\lambda & 1 \1 & 1 & 6-\lambda\end{array} ight| \& =(8-\lambda)\left(\lambda^2-14 \lambda+47 ight)+(-7+\lambda)+(-9+\lambda) \& =(8-\lambda)\left(\lambda^2-14 \lambda+47-2 ight) \& =(8-\lambda)(\lambda-9)(\lambda-5) .\end{aligned}[/latex]

Thus the principal moments are 40ma², 64ma² and 72ma². As a partial check: 40 + 64 + 72 = 8(8 + 8 + 6).

If the angular velocity ω and the angular momentum J = [latex]I[/latex]ω are parallel, then the body is rotating about one of its principal axes (the eigenvectors of [latex]I[/latex]); their ratio is the principal moment about that axis and is thus one of the three values calculated above.

Question 26.17: A rigid body consists of eight particles, each of mass m, he...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions