Rigid bodies - general theorybased on FW-26

Consider a system on N particles with all their relative separations fixed: it has 3 translational and 3 rotational degrees of freedom.Motion with One Arbitrary Fixed Point:

body-fixed frame

inertial frame

= 0

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Kinetic Energy:

continuum limit:

it is convenient to define the inertia tensor:

the kinetic energy has a simple form: in a body-fixed frame it is constant and depends only on the mass distribution

both I and ω depend on the choice of the coordinate system, but T does not!

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Angular momentum:

continuum limit:

and it is related to the kinetic energy:

using the inertia tensor it can be written as:

I, ω and L depend on the choice of the coordinate system!

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General motion with No Fixed Point:

body-fixed frame with the origin at the center of mass

inertial frame

a parallel copy of the inertial frame with the same origin as the body fixed frame - the center-of-mass frame

(in general it is not an inertial frame)

0 for rigid bodies

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Kinetic Energy and Angular Momentum:

body-fixed frame with the origin at the center of mass

inertial frame

a parallel copy of the inertial frame with the same origin as the body fixed frame - the center-of-mass frame

(in general it is not an inertial frame)

Review:

internal motion about the CM

Formulas for T’ and L’ are identical to the case of a motion with one fixed point!

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Then the inertia tensor in a body-fixed frame with the origin displaced by a is given by:

Inertia tensorbased on FW-26

We have to learn how to evaluate the inertia tensor in a body-fixed frame. First, let’s define the inertia tensor in a body-fixed frame with the origin at the center of mass:

parallel-axis theorem!

center-of-mass relations

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Example (a uniform disk of radius a):

moment of inertia about a perpendicular axis through the center of the disk (easy to calculate in cylindrical polar coordinates)

moment of inertia about a perpendicular axis through the edge of the disk (would be tedious to calculate directly)

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Diagonalizing a real symmetric matrix:

Principal axes (a coordinate system in which the inertia tensor is diagonal):

all the formulas simplify in such a coordinate system

similar to solving for normal modes, v I and m 1non-trivial solution only if:

we get 3eigenvalues, 3 eigenvectors, and we can form the modal matrix:

new orthonormal basis - principal axes

T

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Modal matrix diagonalizes the inertia tensor:

we can define a new set of angular velocities:

T

Kinetic Energy becomes:

projections of ω along the principal axes

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Angular Momentum in the new coordinate system:

Principal moments of inertia:

projections of r along principal axes

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Euler’s Equationsbased on FW-27Describing rotational motion:

in terms of body-fixed observables:

valid in any inertial frame but also in the cm frame:

Euler’s equations:

projecting onto the principal body axes

simple formulas, but limited use, since both angular velocities and torques are evaluated in a time dependent body-fixed frame (principal axes);

used mostly for torque-free motion or partially constrained motion (examples follow)

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Applications - Compound Pendulumbased on FW-28a rigid body constrained to rotate about a fixed axis:

stationary both in the inertial frame and in body-fixed frame

we can choose the coordinate system so that:

one degree of freedom

perp. distance from the 3rd axis

in both inertial and body frame

ignoring friction, the only torque comes from gravity:

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for small oscillations, the motion is simple harmonic:

equation of motion of an arbitrary compound pendulum:

Let’s rewrite it in terms of the moment of inertia about the axis going through the center of mass:

radius of gyrationequivalent radial distance of a point mass M leading to the same moment of inertia

radius of gyration about the axis through Q

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oscillation frequency about the axis through Q:

center of percussion

length of a simple pendulum with the same frequency

invert the pendulum and suspend it from the parallel axis going through P

The oscillations frequencies and periods about the axis through Q and P are identical:

is used to measure g!

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Response of a baseball bat to a transverse force: (we neglect gravity)

applied force

reaction force at the point of support

motion of the center of mass (N-2nd):

the only direction R can move due to constraints

torque equation:

if the force is applied at the center of percussion the reaction force at the point of support vanishes!

if applied beyond that point, the reaction force has the same direction as applied force

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Rolling and Sliding Billiard Ballbased on FW-28

a ball is struck at the center (h=0) with a horizontal force. When does it start rolling?

force equation:

initial conditions: x = 0, x = v , ϕ = 0, ϕ = 0

solution:

..0

torque equation:

a�̇ =5

2µgt

Pure rolling without sliding occurs only for:

at this time:

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Rolling and Sliding Billiard Ballbased on FW-28

a ball is struck at h above the center with a horizontal force. When does it start rolling?

initial conditions:

the same equations of motion:

initial angular velocity

independent of the impulse!

Pure rolling without sliding occurs only for:

rolls immediately, no sliding (friction)

slides, then rolls

rolls too fast?!

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