Chapter Questions
Starting from the definition of the position vector of the center of mass, show that$$\sum_{k=1}^n m_k\left(\mathbf{r}_k-\mathbf{r}\right)=\mathbf{0}, \quad \sum_{k=1}^n m_k\left(\mathbf{v}_k-\mathbf{v}\right)=\mathbf{0}$$Where were these identities used?
Starting from the definition of the angular momentum of a system of particles relative to a point $P$, prove that$$\mathbf{H}_P=\left(\mathbf{r}-\mathbf{r}_P\right) \times m \mathbf{v}+\mathbf{H}_C$$
Starting from the definition of the kinetic energy $T$ of a system of particles, show that$$T=\frac{1}{2} m \mathbf{v} \cdot \mathbf{v}+\frac{1}{2} \sum_{k=1}^n m_k\left(\mathbf{v}_k-\mathbf{v}\right) \cdot\left(\mathbf{v}_k-\mathbf{v}\right)$$Using this result, show that the kinetic energy of a system of particles is not, in general, equal to the kinetic energy of the center of mass.
Consider two particles that are free to move on a horizontal surface $z=$ 0 . Vertical gravitational forces $-m_1 g \mathbf{E}_z$ and $-m_2 g \mathbf{E}_z$ act on the respective particles. The position vectors of the particles are$$\mathbf{r}_1=x \mathbf{E}_x+y \mathbf{E}_y, \quad \mathbf{r}_2=\mathbf{r}_1+r \mathbf{e}_r$$Derive an expression for the position vector $\mathbf{r}$ of the center of mass $C$ of this system of particles. Verify your answer by examining the limiting cases that $m_1$ is much larger than $m_2$ and vice versa.
Consider the system of particles discussed in Exercise 7.4. Suppose the particles are connected by a linear spring of stiffness $K$ and unstretched length $L$. Show that the linear momenta $\mathbf{G} \cdot \mathbf{E}_x$ and $\mathbf{G} \cdot \mathbf{E}_y$ are conserved. What do these results imply about the motion of the center of mass $C$ of this system of particles?
For the system of particles discussed in Exercise 7.5, prove that $\mathbf{H}_C \cdot \mathbf{E}_z$ is conserved. What does this result imply about $\dot{\theta}$ ?
Consider the system of particles discussed in Exercise 7.5. Starting from the work-energy theorem, prove that the total energy $E$ of the system of particles is conserved. Here,$$E=\frac{1}{2}\left(m_1 \mathbf{v}_1 \cdot \mathbf{v}_1+m_2 \mathbf{v}_2 \cdot \mathbf{v}_2\right)+\frac{K}{2}(r-L)^2$$
For the cart and pendulum system discussed in Section 7.5, show that $\mathbf{G} \cdot \mathbf{E}_x$ and the total energy $E$ are still conserved if the spring is replaced by an inextensible string of length $L$.
Consider the system of four particles discussed in Section 7.7. If one had the ability to measure $r_1, r_2, r_3, r_4$, and $\omega$ for this system, how would one verify that $\mathbf{H}_O \cdot \mathbf{E}_z$ was conserved?
Referring to the system of four particles discussed in Section 7.7, what are the $\mathbf{e}_{r_i}$ and $\mathbf{e}_{\theta_i}$ components of the balances of linear momenta for each of the four particles? How could the resulting differential equations and conservation of $\mathbf{H}_O$ be used to compute the motions of the four particles (cf. Figure 7.5)?