Question

4. Recall that the angular momentum about the origin of a particle of mass $m$, velocity $v$, and position $\mathbf{r}$ is $\mathbf{J} = \mathbf{r} \times m\mathbf{v}$. Now consider a rigid collection of particles (or a solid body) rotating about an axis through the origin, the angular velocity of the collection is given by $\mathbf{\omega}$. (a) Show that the velocity of the $i^{th}$ particle is $\mathbf{v}_i = \mathbf{\omega} \times \mathbf{r}_i$. (b) Show that the total angular momentum $\mathbf{J}$ is $\mathbf{J} = \sum_i m_i \left[ \mathbf{\omega} r_i^2 - (\mathbf{r}_i \cdot \mathbf{\omega}) \mathbf{r}_i \right]$. To do this, you may want to use the vector identity $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$. (c) Show further that the component of $\mathbf{J}$ along the axis of rotation can be written as $I\omega$, where $I$, the moment of inertia of the collection about the axis or rotation, is given by $I = \sum_i m_i \rho_i^2$. Explain in words what $\rho_i$ is. (d) Using the expression for $I$ above, show that the total kinetic energy of the particles (or solid object) is $\frac{1}{2}I\omega^2$.

          4. Recall that the angular momentum about the origin of a particle of mass $m$, velocity $v$, and
position $\mathbf{r}$ is $\mathbf{J} = \mathbf{r} \times m\mathbf{v}$. Now consider a rigid collection of particles (or a solid body) rotating
about an axis through the origin, the angular velocity of the collection is given by $\mathbf{\omega}$.
(a) Show that the velocity of the $i^{th}$ particle is $\mathbf{v}_i = \mathbf{\omega} \times \mathbf{r}_i$.
(b) Show that the total angular momentum $\mathbf{J}$ is
$\mathbf{J} = \sum_i m_i \left[ \mathbf{\omega} r_i^2 - (\mathbf{r}_i \cdot \mathbf{\omega}) \mathbf{r}_i \right]$.
To do this, you may want to use the vector identity $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{a} \cdot \mathbf{b})\mathbf{c}$.
(c) Show further that the component of $\mathbf{J}$ along the axis of rotation can be written as $I\omega$,
where $I$, the moment of inertia of the collection about the axis or rotation, is given by
$I = \sum_i m_i \rho_i^2$.
Explain in words what $\rho_i$ is.
(d) Using the expression for $I$ above, show that the total kinetic energy of the particles (or
solid object) is $\frac{1}{2}I\omega^2$.

4. Recall that the angular momentum about the origin of a particle of mass m, velocity v, and
position 𝐫 is 𝐉 = 𝐫× m𝐯. Now consider a rigid collection of particles (or a solid body) rotating
about an axis through the origin, the angular velocity of the collection is given by ω.
(a) Show that the velocity of the i^th particle is 𝐯i = ω×𝐫i.
(b) Show that the total angular momentum 𝐉 is
𝐉 = mi [ ω ri^2 - (𝐫i ·ω) 𝐫i ].
To do this, you may want to use the vector identity 𝐚× (𝐛×𝐜) = (𝐚·𝐜)𝐛 - (𝐚·𝐛)𝐜.
(c) Show further that the component of 𝐉 along the axis of rotation can be written as Iω,
where I, the moment of inertia of the collection about the axis or rotation, is given by
I = mi ^2.
Explain in words what is.
(d) Using the expression for I above, show that the total kinetic energy of the particles (or
solid object) is (1)/(2)Iω^2.

Added by Chris M.

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