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Chapter 10: Problem 9

Show that the total angular momentum \(\mathbf{L}\) of a system of particles may be written as $$ \mathbf{L}=M \mathbf{R} \times \dot{\mathbf{R}}+\mathbf{L}_{c} $$ where \(\mathbf{R}\) is the position vector of the centre-of-mass, \(M\) is the total mass of the system and \(\mathbf{L}_{c}\) is the angular momentum of the system relative to the centre-of-mass.

Short Answer

Expert verified
The total angular momentum \(\mathbf{L}\) is expressed as \(M \mathbf{R} \times \dot{\mathbf{R}} + \mathbf{L}_{c}\).

Step by step solution

01

Definition of Total Angular Momentum

The total angular momentum \(\mathbf{L}\) of a system of particles is given by \(\mathbf{L} = \sum_{i} \mathbf{r}_i \times \mathbf{p}_i\), where \(\mathbf{r}_i\) is the position vector of particle \(i\) and \(\mathbf{p}_i = m_i \mathbf{v}_i\) is its linear momentum.
02

Decompose Particle Position Vectors

Each position vector \(\mathbf{r}_i\) can be expressed in terms of the center-of-mass position \(\mathbf{R}\) as \(\mathbf{r}_i = \mathbf{R} + \mathbf{r}_{i,c}\), where \(\mathbf{r}_{i,c}\) is the position of particle \(i\) relative to the center-of-mass.
03

Substitute the Decomposed Vectors into \(\mathbf{L}\)

Substitute \(\mathbf{r}_i = \mathbf{R} + \mathbf{r}_{i,c}\) into the expression for \(\mathbf{L}\). This gives \(\mathbf{L} = \sum_{i} (\mathbf{R} + \mathbf{r}_{i,c}) \times m_i \mathbf{v}_i\).
04

Expand the Cross Product

Expand the expression from the previous step to get \(\mathbf{L} = \sum_{i} (\mathbf{R} \times m_i \mathbf{v}_i + \mathbf{r}_{i,c} \times m_i \mathbf{v}_i)\). This separates into two sums.
05

Simplify the First Term

The first term \(\sum_{i} \mathbf{R} \times m_i \mathbf{v}_i = \mathbf{R} \times \sum_{i} m_i \mathbf{v}_i\) simplifies to \(M \mathbf{R} \times \dot{\mathbf{R}}\) because \(\dot{\mathbf{R}} = \frac{1}{M} \sum_{i} m_i \mathbf{v}_i\) and \(M\) is the total mass.
06

Define Angular Momentum Relative to Center-of-Mass

The second term \(\sum_{i} \mathbf{r}_{i,c} \times m_i \mathbf{v}_i\) is defined as the angular momentum \(\mathbf{L}_{c}\) relative to the center-of-mass.
07

Combine Results

Combine the simplified expressions to show that \(\mathbf{L} = M \mathbf{R} \times \dot{\mathbf{R}} + \mathbf{L}_{c}\), which confirms the given equation for total angular momentum in terms of the center-of-mass and relative motion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center of Mass
The center of mass is an essential concept when studying the dynamics of systems of particles. Imagine it as the "balance point" of a system, where the mass can be thought to be concentrated. This point helps simplify the analysis of complex systems. For a system made up of multiple particles, the center of mass \(\mathbf{R}\) is calculated using the weighted average of their individual positions, given by: \[\mathbf{R} = \frac{1}{M} \sum_{i} m_i \mathbf{r}_i\] where \(m_i\) is the mass of each particle \(i\) and \(\mathbf{r}_i\) is the position vector The term \(M\) represents the total mass of the system, defined as \(M = \sum_{i} m_i\). This concept allows us to simplify the analysis of motion since, under certain conditions, the entire system can be treated as if all mass is located at the center of mass.
System of Particles
Understanding a system of particles involves examining how multiple masses interact and move. Unlike a single particle, where the focus is solely on its path and forces acting on it, a system includes interactions among different particles. This includes gravitational, electromagnetic, and other forces that might be at play. When dealing with a system of particles, we must consider various aspects:
  • Each particle's position, velocity, and acceleration
  • The forces acting on each particle
  • Total mass and the center of mass of the system
  • Conservation laws, such as conservation of momentum
These factors contribute to the overall study of the system's behavior. By analyzing all particles together, we can predict the system's movement and response to forces.
Linear Momentum
Linear momentum is a crucial concept in physics that describes the quantity of motion of a particle. For a single particle, linear momentum \(\mathbf{p}\) is the product of its mass \(m\) and its velocity \(\mathbf{v}\): \[\mathbf{p} = m \mathbf{v}\] For a system of particles, the total linear momentum \(\sum_{i} \mathbf{p}_i\) reflects the combined effect of the momenta of all the particles. The total linear momentum of a system of particles can be related to the motion of its center of mass: \[\sum_{i} \mathbf{p}_i = M \dot{\mathbf{R}}\] where \(\dot{\mathbf{R}}\) is the velocity of the center of mass. Linear momentum is conserved in isolated systems, meaning if no external forces are acting on a system, its total linear momentum remains constant. This is fundamental in solving many problems related to mechanics and dynamics.
Vector Decomposition
To understand complex movements and forces, it is often helpful to decompose vectors. In our context, vector decomposition means expressing a vector with respect to different bases. For example, the position vector of a particle within a system can be decomposed into parts relating to the center of mass and its position relative to the center of mass:\[\mathbf{r}_i = \mathbf{R} + \mathbf{r}_{i,c}\] Here, \(\mathbf{R}\) is the position vector of the center of mass, and \(\mathbf{r}_{i,c}\) is the vector from the center of mass to the particle. This decomposition is crucial when calculating quantities like angular momentum. It allows us to separate the motion of the system as a whole from the motion of the particles within the system.Such decomposition simplifies the mathematical treatment of systems, making it easier to apply the conservation laws and analyze motions separately.

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Most popular questions from this chapter

A child's hoop of mass \(m\) and radius \(r\) rolls without slipping in a straight line with speed \(v\). To change the direction of the hoop, the child taps it with a stick, applying an impulse \(\Delta \mathbf{p}\) that is perpendicular to the plane of the hoop. Where on the hoop should the child apply the blow? Show that the hoop is deflected by an angle $$ \theta \approx \frac{\Delta p}{m v} . $$ You should assume that the plane of the hoop remains vertical throughout.

(a) Show that the moment of inertia tensor of a uniform solid right circular cone about its apex is given by $$ \mathbf{I}=\left(\begin{array}{ccc} \frac{3}{3} m\left[\frac{R^{2}}{4}+h^{2}\right] & 0 & 0 \\ 0 & \frac{3}{5} m\left[\frac{R^{2}}{4}+h^{2}\right] & 0 \\ 0 & 0 & \frac{3}{10} m R^{2} \end{array}\right) $$ where \(m\) is the mass of the cone, \(R\) is the radius of the base, \(h\) is the height and the symmetry axis lies along \(\mathbf{e}_{3}\). (b) A solid right circular cone rolls on its side without slipping on a horizontal surface. The cone returns periodically to its starting position with constant angular speed \(\omega\). Show that the kinetic energy of the cone is given by $$ T=\frac{3 m \omega^{2} h^{2}}{40}\left(\frac{6 h^{2}+R^{2}}{h^{2}+R^{2}}\right) $$

A solid uniform cylinder of density \(\rho\) has radius \(R\) and height \(h\). Use symmetry to deduce the principal axes for rotations about the centre-of-mass. Calculate the corresponding principal moments of inertia. A cylindrical artillery shell of radius \(0.1 \mathrm{~m}\) and length \(0.4 \mathrm{~m}\) is fired from a gun into the air. Estimate the rate of precession of the symmetry axis if the barrel of the gun imparts a spin of 50 revolutions per second about the symmetry axis of the shell.

A rotating thin circular disc, moving through a fluid, is subject to a damping torque about its centre of mass that is given by $$ \begin{aligned} &\tau_{1}=-\kappa \omega_{1} \\ &\tau_{2}=-\kappa \omega_{2} \\ &\tau_{3}=0 \end{aligned} $$ where \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) are the principal axes that lie in the plane of the disc, \(\mathbf{e}_{3}\) is the symmetry axis and \(\omega\) is the angular velocity. Use Euler's equations with the substitution \(\eta=\omega_{1}+i \omega_{2}\) to determine the time-dependence of \(\omega_{1}^{2}+\omega_{2}^{2}\). Describe the motion of the disc in the lab frame.

A coin may be rolled without slipping in a circular path on a horizontal surface, as long as it leans slightly towards the centre of the circle. Show that the angle of tilt \(\phi\) of the coin is given by $$ \tan \phi \approx \frac{3 v^{2}}{2 g R}, $$ where \(v\) is the constant speed of the coin, \(R\) is the radius of the circle and \(g\) is the gravitational acceleration. What happens if \(v\) decreases slowly with time due to rolling friction?
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