Systems of particles and angular momentum

The last three lessons treated one particle at a time. Real machines are made of many parts that push and pull on each other, and many important problems involve a stream of material rather than a single body: sand pouring onto a belt, air passing through a fan, water leaving a nozzle. This lesson extends the three kinetic principles to systems of particles, introduces angular momentum, and applies both to steady streams of mass. It is also the bridge to rigid bodies, which are simply systems of particles that keep fixed distances from each other. Almost every result in Lessons 10 and 11 is a special case of something in this lesson.

The center of mass and its motion

For a system of particles with masses \(m_i\) at positions \(\mathbf{r}_i\), the center of mass \(G\) is the mass-weighted average position, \(\mathbf{r}_G = \sum m_i \mathbf{r}_i / m\), where \(m\) is the total mass. Writing the second law for each particle and adding them up, the internal forces cancel in pairs by Newton's third law, leaving

This is a remarkable result. However the parts of a system move relative to each other, its center of mass moves as if all the mass were concentrated there and all the external forces acted there. A tote whose contents shift during braking, a shuttle carrying a loose load, a two-stage telescoping fork: their centers of mass all obey this one equation, and the internal forces between the parts never enter. The equation does not tell you how each part moves, but it often tells you what you need: the total external force required, or where the system as a whole will end up.

A useful consequence: if the resultant external force on a system is zero in some direction, the center of mass does not accelerate in that direction. If it starts at rest, it stays put. When a person walks forward on a small boat, the boat moves backward so that the center of mass of person and boat stays where it was. When a shuttle's telescoping forks extend sideways on a frictionless rail, the shuttle body shifts the other way. On a real rail, friction is an external force and limits the effect, but on air bearings or well-lubricated slides it can surprise a designer.

Work, energy and momentum for systems

The impulse-momentum principle carries over directly: the total linear momentum of a system changes only by the impulse of the external forces, because internal impulses cancel. Work and energy is more subtle. Internal forces act in equal and opposite pairs, but the points where they act can move different distances, so their work does not necessarily cancel. Two blocks joined by a stretched spring, a tote sliding on a shuttle deck, and parts rubbing in a clutch all have internal forces that do net work, either storing energy in a spring or dissipating it as heat. For a system, the work-energy principle must therefore include the work of internal forces as well as external ones:

For rigid bodies the internal forces between particles do no net work, because the particles keep fixed distances. That is one reason rigid-body energy methods (Lesson 11) are so clean. For systems with springs, friction between parts or deformation, the internal work is real and must be accounted for, as the collision examples of Lesson 6 showed.

Angular momentum of a particle

The angular momentum of a particle about a fixed point \(O\) is the moment of its linear momentum about that point:

In plane motion it is a vector perpendicular to the plane, with magnitude \(m v d\), where \(d\) is the perpendicular distance from \(O\) to the line of the velocity. Taking the time derivative and using the second law shows that the resultant moment of the forces about \(O\) equals the rate of change of angular momentum about \(O\):

Integrating over time gives the principle of angular impulse and momentum, \((\mathbf{H}_O)_1 + \sum\int \mathbf{M}_O\,dt = (\mathbf{H}_O)_2\). And if the resultant moment about \(O\) is zero, angular momentum about \(O\) is conserved. This happens whenever every force on the particle passes through \(O\), such as a cord pulling toward a hub or gravity pulling a satellite toward the center of the Earth. Such forces are called central forces.

In polar coordinates with \(O\) at the origin, the angular momentum has magnitude \(m r^2\dot\theta\). Conservation therefore means that \(r^2\dot\theta\) stays constant: as the particle moves closer to \(O\), its angular velocity increases with the inverse square of the distance. Kepler noticed this in planetary orbits four centuries ago; figure skaters exploit it whenever they pull their arms in to spin faster.

The example shows that angular momentum and energy are separate accounts. The program below checks the result by integrating the polar equations of motion while the cord pulls the slider in at a steady 0.20 m/s. It confirms that \(m r^2 \dot\theta\) stays constant to four figures and that the work done by the cord tension matches the gain in kinetic energy. The small extra kinetic energy from the radial motion itself, \(\tfrac12 m\dot r^2\), is the same at start and end, so it cancels.

# A 0.50 kg slider on a light, freely spinning rod is pulled toward the hub
# by a cord at a steady 0.20 m/s. No torque acts about the hub, so
# angular momentum H = m r^2 theta_dot should stay constant. Integrate the
# polar equation for theta and check H, then compare the cord's work with
# the change in kinetic energy.
m, r, w = 0.50, 0.60, 2.0          # kg, m, rad/s
rdot = -0.20                        # m/s (cord pulls the slider in)
dt, t = 1e-5, 0.0
work_cord = 0.0
T0 = 0.5 * m * ((r * w)**2 + rdot**2)
H0 = m * r * r * w

while r > 0.30:
    # transverse equation with F_theta = 0:  r*wdot + 2*rdot*w = 0
    wdot = -2 * rdot * w / r
    # radial equation gives the cord tension: -T = m (rddot - r w^2), rddot = 0
    T = m * r * w * w
    work_cord += T * (-rdot) * dt   # tension points inward, slider moves inward
    w += wdot * dt
    r += rdot * dt
    t += dt

T1 = 0.5 * m * ((r * w)**2 + rdot**2)
print(f"after {t:.2f} s: r = {r:.3f} m, omega = {w:.3f} rad/s")
print(f"angular momentum: start {H0:.4f}, end {m * r * r * w:.4f} kg m^2/s")
print(f"kinetic energy: start {T0:.3f} J, end {T1:.3f} J, change {T1 - T0:.3f} J")
print(f"work done by the cord: {work_cord:.3f} J")

after 1.50 s: r = 0.300 m, omega = 8.000 rad/s
angular momentum: start 0.3600, end 0.3600 kg m^2/s
kinetic energy: start 0.370 J, end 1.450 J, change 1.080 J
work done by the cord: 1.080 J

The same effect appears in machines. A robot that retracts its arm while slewing freely will speed up unless its drive applies a braking torque; a centrifugal clutch whose shoes fly outward slows the spin of the parts that carry them. Lesson 11 extends conservation of angular momentum to rigid bodies, where the mass moment of inertia takes the place of \(m r^2\).

Steady streams of mass

Many machines move a continuous stream of material: belts carrying bulk solids, pumps and nozzles moving liquid, fans moving air. Applying impulse and momentum to a fixed region of space, a control volume, through which mass flows at a steady rate \(\dot m\), gives

The resultant force on the material in the control volume equals the mass flow rate times the change in velocity between where it enters and where it leaves. This is how the thrust of a jet, the force on a pipe bend and the reaction on a fire hose nozzle are found. It also gives the force a conveyor needs to accept material.

The factor of one half is not a quirk of this example. Whenever material is accelerated to a belt's speed by sliding friction, exactly half of the work done ends up as heat, whatever the friction coefficient. It is the continuous version of the perfectly plastic impact in Lesson 6. Conveyor designers reduce the loss, and the wear and dust that come with it, by feeding material in the direction of belt travel at close to belt speed, using curved chutes or accelerator belts. That is the same idea as matching speeds at a parcel transfer.