Lecture II: Rigid-Body Physics
2Lecture II: Rigid-Body Physics
Previously: Point dimensionless objects moving through a trajectory.
Today: Objects with dimensions, moving as one piece.
3Lecture II: Rigid-Body Physics
Rigid-Body Kinematics Objects as sets of points. Relative distances between all points are
invariant to rigid movement. Free body movement: around the center of mass
(COM). Movement has two components:
Linear trajectory of a central point. Relative rotation around the point.
4Lecture II: Rigid-Body Physics
Mass The measure of the amount of matter in the
volume of an object:
: the density of each point the object volume . : the volume element.
Equivalently: a measure of resistance to motion or change in motion.
5Lecture II: Rigid-Body Physics
Mass For a 3D object, mass is the integral over its
= ###(, , )
For uniform density ( constant):
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Center of Mass The center of mass (COM) is the average point
of the object, weighted by density:
( : point coordinates.
Point of balance for the object. Uniform density: COM centroid.
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Center of Mass of System Sets of bodies have a mutual center of mass:
8:; 8: mass of each body. 8: location of individual COM. = 898:; .
Example: two spheres in 1D
=>? =;; +AA; +A
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Center of Mass
Quite easy to determine for primitive shapes
What about complex surface based models?
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Rotational Motion a point on to the object. is the center of rotation. Distance vector = .
= : the distance. Object rotates
travels along a circular path. Unit-length axis of rotation: .
Here, = (out from the screen). Rotation: counterclockwise.
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Angular Displacement Point covers linear distance .
is the angular displacement of the object:
: arc length.
Unit is radian ()
1 radian = angle for arc length 1 at a distance 1.
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Angular Velocity Angular speed: the rate of change of the angular
unit is UVW XYZ .
The angular velocity vector is collinear with the rotation axis:
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Angular Acceleration Angular acceleration: the rate of change of the
Paralleling definition of linear acceleration.
Unit is /A
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Tangential and Angular Velocities Every point moves with the same angular velocity.
Direction of vector: .
Tangential velocity vector:
= ^_ U (abs. values) due to = 4 .
Only the tangential part matters!
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Dynamics The centripetal force creates curved
In the direction of (negative) Object is in orbit.
Constant force circular rotation with constant tangential velocity. Why?
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Tangential & Centripetal Accelerations Tangential acceleration holds:
cf. velocity equation = .
The centripetal acceleration drives the rotational movement:
U = A.
What is the centrifugal force?
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Angular Momentum Linear motion linear momentum: = . Rotational motion angular momentum about
any fixed relative point (to which is measured):
= # (
unit is 4 4 = (mass element)
Angular momentum is conserved! Just like the linear momentum.
Caveat: conserved w.r.t. the same point.
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Angular Momentum Plugging in angular velocity:
Integrating, we get:
Note: The angular momentum and the angular velocity are not generally collinear!
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Moment of Inertia Define: =
For a single rotating body: the angular velocity is constant. We get:
A + A f g hf + A + A g hf g + (A + A)h
ff fg fhgf gg ghhf hg hh
Note: replacing integral with a (constant) matrix operating on a vector!
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Momentum and Inertia The inertia tensor only depends on the geometry
of the object and the relative fixed point (often, COM):
fg = gf = #
fh = hf = #
gh = hg = #
ff = # A + A
gg = # A + A
hh = # A + A
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The Inertia Tensor Compact form:
# A + A # #
# # A + A #
# # # A + A
The diagonal elements are called the (principal) moment of inertia.
The off-diagonal elements are called products of inertia.
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The Inertia Tensor Equivalently, we separate mass elements to
density and volume elements:
= # , , A + A A + A A + A
The diagonal elements: distances to the respective principal axes.
The non-diagonal elements: products of the perpendicular distances to the respective planes.
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Moment of Inertia The moment of inertia j, with respect to a
rotation axis , measures how much the mass spreads out:
j = # j A(
j: perpendicular distance to axis. Through the central rotation origin point.
Measures ability to resist change in rotational motion. The angular equivalent to mass!
23Lecture II: Rigid-Body Physics
Moment and Tensor We have: j A = A = k() for any point . (Remember: is distance to origin).
Thus:j = k()
The scalar angular momentum around the axis is then j = j.
Reducible to a planar problem (axis as axis).
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Moment of Inertia For a mass point:
= 4 jA
For a collection of mass points: = 88A8
For a continuous mass distribution on the plane: = jA?
A Ao o
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Inertia of Primitive Shapes
For primitive shapes, the inertia can be expressed with the parameters of the shape
Illustration on a solid sphere Calculating inertia by integration of
thin discs along one axis (e.g. ). Surface equation: A + A + A = A
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Inertia of Primitive Shapes Distance to axis of rotation is the radius of the disc at the
cross section along : A = A + A = A A.
Summing moments of inertia of small cylinders of inertia q= U
along the z-axis:
q =12A =
q =;A uvwv =
;A A A Avwv =
;A[u 2A o 3
+ z 5 ]wvv = 1 2 3 + 1 5 z.
As = 4 3 o, we finally obtain: q =AzA.
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Inertia of Primitive Shapes Solid sphere, radius and mass :
Hollow sphere, radius and mass :
A 0 0
A 0 0
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Inertia of Primitive Shapes Solid ellipsoid, semi-axes , , and mass :
Solid box, width , height , depth and mass :
A+A) 0 0
112(A+A) 0 0
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Inertia of Primitive Shapes Solid cylinder, radius , height and mass :
Hollow cylinder, radius , height and mass :
A+A) 0 0
A+A) 0 0
0 0 A
30Lecture II: Rigid-Body Physics
Parallel-Axis Theorem The object does not necessarily rotate around the
center of mass. Some point can be fixed!
parallel axis theorem:
= =>? +A
^: inertia around axis . =>? inertia about a parallel axis through the COM. is the distance between the axes.
31Lecture II: Rigid-Body Physics
Parallel-Axis Theorem More generally, for point displacements:
f, g, h
ff = # A + A + fA
gg = # A + A +gA
hh = # A + A +hA
fg = # + fg
fh = # +fh
gh = # +gh
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Perpendicular-Axis Theorem For a planar 2D object, the moment of inertia
about an axis perpendicular to the plane is the sum of the moments of inertia of two perpendicular axes through the same point in the plane:
h = f + gfor any planar object
h = 2f = 2gfor symmetrical objects
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Reference Frame The representation of the inertia tensor is coordinate
The physical effect should be invariant to coordinates!
If transformation changes bases from body to world coordinate, the inertia tensor in world space is:
UW = 4 Wg 4 k
The moment of inertia is invariant! Why?