6/06 Rotational Dynamics

Rotational Dynamics

About this lab: In certain circumstances (only action-reaction third law “internal” forces), Newton's laws predict a conserved (unchanged) quantity, total angular momentum, for a system of interacting parts.

This is true independent of force type or details of internal interaction, offering great simplification if the assumptions are satisfied.

Angular momentum is a vector quantity, so attention to direction and signs is important.

The existence of the angular momentum conservation law is related to rotational coordinate invariance, the independence of the laws of physics from angular orientation. (This does not mean that no regions vary in orientation properties, just that physical law does not vary.)

If there are external forces (non action-reaction Third Law pair forces), the angular acceleration of the system as a whole changes in proportion to the net external torque. For a rigid body, the constant of proportionality is the moment of inertia, I.

Apparatus: Pasco rotary motion sensor, Vernier LabPro interface, Vernier Logger Pro data acquisition programs (2), aluminum disk, long rod, string, weights.

References: Cutnell & Johnson: Physics, 6th Chapters 8 and 9.Serway & Beichner: 5th, v1 Examples 10.12, 11.9, 11.10, Fig 11.12

Introduction Rotating systems involve concepts that are similar to those in translational motion. The magnitude of rotational or angular velocity is given by

ω = v/r (1)

where v is the velocity that is perpendicular to the radius, r , from the center of rotation. As with linear velocity, angular velocity is a vector, but a more complicated “cross-product” vector formed from the two vectors r and v . We will measure only magnitudes.

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“Inertia” is the generic word for the “resistance” an object has to being accelerated. It is called the “moment of inertia” for rotational acceleration and depends upon how the masses of the rotating body are distributed and on the location of the axis of rotation with respect to the center of mass. In the case of a single, compact mass particle, m , moving in a circle of radius, r, the moment of inertia is

I m = mr2 (2)

For a solid disk the moment of inertia about a perpendicular axis through its center of mass, where mass=M and radius=R, is determined by adding the contributions of each particle in the disk according to their perpendicular distance to the axis. The net result is:

I disk = ½ mR2

Rotational Energy For translational motion the kinetic energy is defined as

12

mvcm2 where v is the linear velocity of the center of mass (c.m.). Rotational kinetic

energy is

KE rotational = 12

I 2

where is the angular velocity about the rotation axis. The total mechanical energy of a rotating and translating rigid object of mass, m, is the sum of translational kinetic energy, rotational kinetic energy, and translational potential energies. Relative to the center of mass, c.m. and an axis of rotation through the c.m. :

E total = KE rotational + KE translational + PE translational

E total =12

mvcm2 + 1

2I 2 + mgh c m (3)

where h is the height (gravitational PE), and the variation of g with height is ignored.

In an earlier experiment we examined the conservation of energy for a bucket (drive mass) falling some distance as the bucket pulled a low friction cart on a track. This demonstrated how the total mechanical energy of translational motion was conserved. The sum of the kinetic and potential energies of the glider and bucket was the same before and after the bucket fell (ignoring friction, pulley and wheel spin energy, and motion of the string). The

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potential energy of the bucket was converted into kinetic energy of the bucket and cart.

Angular Momentum Linear momentum is conserved when objects interact, if there is no net external force. For example, when two gliders collide on an air track they briefly exert equal and opposite forces on each other during the collision. The net (horizontal) force is zero and total momentum is conserved, though there is exchange of individual momentum. With gliders having magnetic bumpers that do not permanently absorb energy (convert organized mechanical energy into heat) the collision is called "elastic”, where both mechanical energy and momentum are conserved. In an inelastic collision, in which the gliders stuck together, momentum is conserved but not mechanical energy; some mechanical energy is lost to deformation. A low friction track experiment demonstrates conservation of translational (linear or straight-line) momentum.

In both translational (straight-line) and rotational motion, momentum is the product of an “inertia” and a “velocity”. In translational cases it is p = mv . Similarly, for its rotational counterpart, called angular momentum L, we have

L = I or simply, L = Iω (4)

L is angular momentum, I is “moment of inertia” and ω is the angular velocity in (rad/sec).

Angular momentum is conceptually similar to translational momentum, but the reference object is the axis of the rotational motion, rather than a point origin.. Consider a swinging ball of mass m, tied to a string of length r. Its moment of inertia is mr2; its angular velocity is v/r. Thus, its angular momentum is Iω = mr2 v/r = mvr, where the length of string defines the radius, r, of the circular motion. Quickly shortening the string to reduce r results in the ball swinging around faster. v increases because r decreases. The angular momentum mvr remains constant. This is the same principle by which a figure skater makes herself spin faster. She gives herself a slow spin on one leg with her arms and the other leg stretched out. Bringing them inwards increases her spinning rate.

Treatment of angular momentum dynamics involves relations analogous to those of linear momentum, in particular

Angular impulse (product of net torque and time of application). This produces a change

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in net angular momentum, which may increase or decrease the initial value, depending on relative directions.

Rotational work (product of torque and angle turned during application). This produces a change in the rotational kinetic energy, which may be an increase or a decrease.

The low friction rotation apparatus shown below, in angular momentum conservation orientation, can be used to study both angular momentum conservation (horizontal disk orientation, without external torques) and total energy conservation without angular momentum conservation (vertical disk orientation, with external torque.) It involves a low friction shaft and angular measurement hardware. On the shaft are mounted: above, an aluminum disk and a toothed plastic rod catcher; below, two low-mass plastic drive wheels. The direction of tooth cut dictates the direction of initial disk rotation.

These cases involve:

Case A) Strong internal torques but no external torques (angular momentum conservation expected, theoretically),

Case B) Constant external torque and no internal torque (constant angular acceleration expected, theoretically).

Pictures below show the apparatus usage for these two investigations: angular momentum conservation and total energy conservation (horizontal and vertical orientations.)

A. Angular Momentum Conservation: No External Torques Acting

Open the Logger Pro file Angular Momentum Conservation.cmbl.

Study the figures, procedures and interpretations outlined below.

Test experimentally angular momentum and energy conservation, for various drop masses and geometries. Redefine dropped object quantities as necessary (I's, m's), and strike through invalid time data. Then examine the graphs. In each case interpret the kinematic and energy graphs regarding:

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Conservation of total energy, conservation of angular momentum, partition of angular momentum among aluminum disk and other masses (theory vs. experiment).

Report A: Print the graphs for at least one good example, as instructed. Record: total energies (initial i, final f, ratio f/i); total angular momenta(i, f, i/f); individual angular momenta (disk, other , experimental ratio other/disk). Give theoretical ratio of individual angular momenta: I other/I disk.). Compare experimental and theoretical angular momenta ratios).

Figure 1 Low friction rotation and angle-counting apparatus, in horizontal, angular-momentum conservation orientation. The small drive pulleys are not used in this orientation.

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Figure 2 Rotation apparatus, with centered flexible shaft in rod form

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Figure 3 Rod in drop position for angular momentum conservation test. There will be strong internal torques, but no external torques. Initial disk angular velocity must be ccw, seen from above. The effect of rod drop on the final common rod-disk angular velocity will depend on the final rod moment of inertia relative to the axis of rotation. If this axis passes through the rod center, I rod c m = 1/12 M rod L2 .

If the rod center of mass is offset from the rotation axis by distance d, a quantity M rod d2 must be added to I rod c m. This is the parallel axis theorem.

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Figure 4 Logger Pro graphical representation of rod-drop collision, with partially invalid data

The sign of theta is dictated by the prescribed sense of rotation which causes the rod to roll down the rod-catcher teeth. (- omega) is plotted to make it and other calculated quantities positive in the graphs (and in data tables).

Rod quantities (L, KE) were calculated assuming that the rod was dropped centered. If this is not true (check!), I rod should be changed in the definitions of L rod and KE rod, in accordance with the parallel axis theorem.

The collision is over by about 1.8 seconds. Rod angular momentum is false before then, should be zero (algorithm assumes rod always spins with disk). Therefore, total energy is also false Total angular momentum appears to be non-conserved after collision. Rod data (L rod and KE rod) prior to 1.8 seconds should be struck through (Edit; Strike Through) in the data table. (This also removes quantities linked in the calculation, i.e. L total and E total before 1.8 seconds. L total = L disk + L rod, E total = E disk + E rod).

The graphs below show the results of these data strikethroughs.

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Figure 5 Same data table as Figure 4 (Logger Pro graphical representation of the rod-drop collision), but with only valid data (i.e. after removal of the false data before 1.8 second drop completion). Compare with the lower quarter of the Figure 4 angular momentum and energy graphs.

The linked quantities L rod and L total, and KE rod and E total are now removed before 1.8 seconds . Only the disk curve remains, which does represent all the angular momentum before collision, and all the kinetic energy.

After the collision, the angular momentum graph (upper right panel) shows the total angular momentum to be largely the same before (green) and after (red top curve) the collision, despite the violent internal torques.

The energy graph (lower right) shows that the collision is quite inelastic – total energy is not conserved at all. Again, only the disk (green) has energy before the collision, and only the disk energy is plotted: E total = E disk, before collision.

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The figure blow indicates an alternative drop configuration with the same mass, but different moment of inertia.

Figure 6 Rotation apparatus with drop mass in approximate circular form. The flex mass is the same as when in rod form, but the mass distribution about the center of mass differs.

For simplicity, we will treat this shape as circular, with mass lumped at the mean radius.

B. Non Conserved Angular Momentum with Total Energy Conservation: External Torques Acting

Close Logger Pro Angular Momentum.cmbl Conservation. Open Logger Pro Constant Torque Rotational Motion.cmbl.

Reorient the rotation apparatus to a vertical disk position. Anchor a thread or string through a pulley slot and wrap a thread or string with small weight suspended around one of the drive pulleys. The string must be wrapped tightly enough not to slip. Then the external torque exerted is mgR pulley (0.025 m for the larger). This being constant (we

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neglect any variation in earth gravitational pull with distance from earth center), the constant angular acceleration equations apply. These have the same form as those for constant force in translational acceleration.

Study the figures, procedures and interpretations outlined below. The drop mass potential energy PE is set to zero by subtraction of the time-zero height. (This is accomplished with the two data columns “theta sub” (subset) and “init theta”.

(This is not a “before and after” collision test. We are interested in the continuous time interval from time zero until dropping drive mass hits. We are not interested in any data after drop mass hits the table. Strike through subsequent data only if it makes you feel better.)

Test experimentally total energy and angular momentum conservation, for various rotating mases (try various additional loadings on disk), drop masses and pulley radii. Examine the graphs. In each case interpret the kinematic, momenta and energy graphs regarding conservation of total energy (?), conservation of total angular momentum (?),

The slope of a linear fit (valid region only) to the plot of vs. t will give the angular acceleration , as will the coefficient of the t2 term in a quadratic fit to the plot of vs. t . Try both and compare. Does appear to be constant, as it should be for constant external torque?

The picture and Logger Pro graph below show the operation and result, for 20 gram drop mass on larger pulley.

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Figure 7 Known external gravitational torque, ready for release.

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Figure 8 Above: Graphical representation of angular motion of disk driven by falling weight, with other calculated quantities.

The 20 gm weight hit the table at about 1.6 seconds; algorithms fail for later data (the PE no longer changes). Most of the drive mass gravitational PE (red) is converted to disk rotational KE (blue); the KE of the falling drive mass is relatively very small (both theoretically and experimentally). Total energy (green) appears to decrease slightly.

Within the valid time region, (all as expected for constant torque angular acceleration ):

> Theta is well fit by a quadratic;> Omega is well fit linearly;

(Angular acceleration α is the (constant) slope: α = dω/dt.)> Disk angular momentum is well fit linearly;> Drive mass PE is well fit by a quadratic.

(Note, drive mass PE would be linear if plotted vs. height h, but this is a time plot.)If the string slips while the drive mass falls, assumptions are invalid!

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Orbital and Spin Angular Momentum and the Periodic Table

If you turn your laboratory, and everything that affects your experiment through some angle, the results of your experiment should not be affected. This symmetry of the laws of physics under rotations also gives rise to a conserved quantity. This conserved quantity is called angular momentum and arises, naturally, in the context of rotational motion of objects”

http://theory.uwinnipeg.ca/mod_tech/node40.html

There are two types of angular momentum: orbital (like the earth's revolution about the sun) and spin (like the earth's rotation on its own axis.

The magnitude of orbital angular momentum depends on the choice of axis about which to calculate it.

But, the magnitude of spin angular momentum is independent of axis choice! There's a difference!

Spin angular momentum is intrinsic; orbital angular momentum is extrinsic.

The state of an electron orbiting a positive nucleus in an atom exhibits both types. The

orbital angular momentum is integer (in units of h

2); the spin (intrinsic angular

momentum) is always 12 in the same units, where h is Planck's quantum constant.

By the action of the Pauli exclusion principle, applied to the electrons in a multi-electron atom, electrons successively fill an s orbit (l = 0), p orbits (l = 1), d orbits (l = 2) etc., and each twice, once with intrinsic spin up and once down. Thus derives the periodic table.

And, the Pauli principle is an expression of relativistic quantum mechanical “statistics” (occupation rules, more exactly wave function properties) for spin ½ half particles like electrons.

Biology is chemistry. (Chemistry is physics.) And the intrinsic spin 1/2 of an electron is the ultimate source of the Pauli principle, and of the chemistry of the periodic table and of our universe.