Assignment 5: Rigid Bodies#

Due: 11:59pm Wednesday 15 November 2023

Rolling Ring#

Consider a ring rolling in a circle. Treat the ring as thin, with radius \(R\) and uniform mass \(M\). Assume that the ring rolls without slipping. Determine all properties of the motion (period of the orbit, radius of the circle, angle of the ring, angular velocities etc.) Be sure to clearly sketch the geometry of your problem, perform a dimensional analysis, and check your solution by considering various limits.

Show how to compute the moment of inertial tensor \(I_{ij}\) from its definition:

\[\begin{gather*} I_{ij} = \int \d{m} (r^2\delta_{ij} - r_i r_j). \end{gather*}\]

Rolling Hoop (Phantom Torques)#

Solve for the equations of motion for a rolling half-circle as discussed in the class notes Phantom Torque and shown below:

Show code cell source Hide code cell source

%%manim -v WARNING -qm RollingHoop
from manim import *
from types import SimpleNamespace


class Hoop:
    R = 4.0
    theta0 = 0.6
    r_cm = 2 / np.pi * R
    m = 1.0
    I_o = m * R ** 2
    I_cm = I_o - m * r_cm ** 2
    shift = UP

    def __init__(self, **kw):
        for k in kw:
            assert hasattr(self, k)
            setattr(self, k, kw.pop(k))
        assert not kw
        
        self.objects = self.annotations

    def get_arc(self, theta=None):
        """Return a (arc, points) for a hoop"""
        color = 'white'
        if theta is None:
            theta = self.theta0
            color = 'BLUE_C'
            
        z_o = -self.R * theta + 0j
        z_cm = z_o - 1j * self.r_cm * np.exp(theta * 1j)
        z_p = z_o - 1j * self.R
        o, p, cm = [np.array([_z.real, _z.imag, 0]) for _z in [z_o, z_p, z_cm]]
        e1 = o + self.R * np.array([np.cos(theta), np.sin(theta), 0])

        arc = Arc(
            radius=self.R, start_angle=PI + theta, angle=PI, arc_center=o,
            color=color
        )
        arc.stroke_width = 10.0
        points = SimpleNamespace(O=o+self.shift, P=p+self.shift, C=cm+self.shift)
        arc.shift(self.shift)
        
        # These are the actual objects we display
        return (arc, points)

    @property
    def annotations(self):
        """Return a `Group` object with the hoop and all of the annotations."""
        arc, points = self.get_arc()
        
        objs = []
        P, C, O = points.P, points.C, points.O
        for _p0, _p1, _t in [
            (arc.get_end(), O, "R"),
            (C, O, r"a"),
        ]:
            objs.append(BraceBetweenPoints(_p0, _p1))
            objs.append(objs[-1].get_tex(_t))
            #objs[-1].font_size = 40
        objs.extend(
            [
                LabeledDot("O", point=O),
                LabeledDot("P", point=P),
                LabeledDot("C", point=C),
            ]
        )
        
        OC = DashedLine(O, C, color='BLUE_C')
        OP = DashedLine(O, P, color='BLUE_C')
        theta = Angle(OP, OC, radius=0.7, dot_radius=2.0)
        theta_l = Tex(r"$\theta$")
        _m = theta.get_midpoint()
        theta_l.next_to(O + 1.5*(_m-O), 0)
        objs.extend([OC, OP, theta, theta_l])

        floor = Line((-6, -self.R, 0), (6, -self.R, 0))
        floor.shift(self.shift)
        floor.stroke_width = arc.stroke_width
        self.arc = arc
        annotations = Group(arc, floor, *objs)
        return annotations


class HoopAnimation(Animation):
    def __init__(self, hoop, **kw):
        super().__init__(hoop.get_arc(theta=hoop.theta0)[0], **kw)
        self.hoop = hoop
        
    def interpolate_mobject(self, alpha):
        # Fake dynamics
        theta = self.hoop.theta0 * np.cos(2*np.pi * alpha)
        self.mobject.points = self.hoop.get_arc(theta=theta)[0].points
        

class RollingHoop(Scene):
    def construct(self):
        hoop = Hoop()
        self.add(hoop.annotations)
        #self.play(HoopAnimation(hoop=hoop), run_time=3, rate_func=linear)

../_images/77f4027af593be4a28d56779ee75b0c4602ac38fab34e8b21023bef8b520f65d.png

Here we have half of a hoop of mass \(m\) and radius \(R\), with center of mass at \(C\) which is distance \(a\) from the center of the hoop \(O\). The external torque is provided by the downward acceleration due to gravity \(g>0\). The hoop rolls without slipping along the floor with point of contact \(P\). The aim of this problem is to obtain the equations of motion for \(\theta(t)\).

Compute the moments of inertia \(I_{O}\), \(I_{C}\), and \(I_{P}(\theta)\) and relate these to each other using the parallel axis theorem. Express your answer in terms of \(m\), \(R\), \(a\), and \(\theta\). (For this problem, show that \(a=2R/\pi\), but express everything in terms of \(a\). This allows you to generalize to symmetric but unequal mass distributions.)
Solve for the equations of motion using the Lagrangian framework about the two points where the rotational and transitional motions can be decoupled. Show that this gives the same equations of motion, with the parallel axis theorem adjusting the moments of inertia exactly as needed to correct the equations.
Solve the equations of motion using Newton’s law about the center-of-mass \(C\):

\[\begin{gather*} \diff{\vect{L}}{t} = \vect{\tau}. \end{gather*}\]

Draw all forces and calculate the torque \(\vect{\tau}\) in terms of these.
Alternative corrections to Newton’s law are discussed in [Jensen, 2011] (available on Perusal for discussion), but I do not find any of these very intuitive. Please let me know if you find any of these intuitive.

Try solving the equations of motion using Newton’s law about the instantaneously stationary point \(P\). You should find that a simple application of Newton’s law does not give the correct answer. Instead, you will need to correct Newton’s law to include a “phantom torque” [Turner and Turner, 2010].

Show that these corrections arise from the \(\theta\) dependence of \(I_{P}(\theta)\). For this reason, one can use \(P\) for the analysis if \(I_{P}\) is independent of \(\theta\) as it is for a complete hoop, balls, cylinders etc.

Nutation#

Using the parametrization of the Euler angles in Landau and Lifshitz Volume 1, describe nutation in the small amplitude limit (i.e. treat this as an application of the normal-mode theory discussed in class.) Clearly state all steps in formulating the problem, i.e. deriving the equations (use the Lagrangian approach, and conserved quantities to obtain an effective 1D theory with an appropriate effective potential), find the “stationary” solution - this corresponds to simple precession. Check this by considering the Newton’s law in the form of the time-evolution of the angular momentum \(\dot{\vec{L}} = \vec{\Omega}\times\vec{L} = \vec{\tau}\). Then, describe the deviations from the stationary solution to derive the normal modes. Try to identify all of the modes shown in Fig. 49, and check your answer numerically using the analytic solution (7). (References are version available on Perusall.) Let me know if you need any help setting up the numerical checks. I recommend using the scipy.integration.quad routine.

Assignment 5: Rigid Bodies

Contents

Assignment 5: Rigid Bodies#

Rolling Ring#

Rolling Hoop (Phantom Torques)#

Nutation#