Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Introduction to Dynamics and Vibrations- Notes, Videos, MCQs

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The document Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev is a part of the Civil Engineering (CE) Course Introduction to Dynamics and Vibrations- Notes, Videos, MCQs.
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6.4.1 Deriving the linear momentum formula

By definition  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev We ca re-write this as follows:

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

(we used the definition of the COM to get the last result)

6.4.2 Deriving the angular momentum formula

Start with the definition:  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Note that ri =rG + di and recall the relative velocity formula vi −vG = ω × (ri− rG) = ω ×di .  This means we can re-write the angular momentum as

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Note that

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Finally, recall the dreaded triple cross product formula

a × b × c= (a ⋅ c)b − (a ⋅b)c

This means that

di × ω× di = (di ⋅ di )ω − di (di ⋅ ω)

We can expand this out

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

This shows that

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Finally collecting terms gives the required answer

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

6.4.3 Deriving the kinetic energy formula

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

We can use vi −vG = ω × (ri− rG) = ω ×di

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Recall that 

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

and expand the dot product of two cross products using the formula

(a × b) ⋅ (c × d) = (a ⋅c)(b ⋅ d) − (b ⋅ c)(a ⋅d)

This shows that 

(ω x di) • (ω x di) = (ω • ω)(di • di) - (w • di)2

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

= ω ⋅IG ω

6.4.5 Calculating the center of mass and inertia of a general rigid body

It is not hard to extend the results for a system of N particles to a general rigid body. We simply regard the body to be made up of an infinite number of vanishingly small particles, and take the limit of the sums as the particle volume goes to zero. The sums all turn into integrals.

3D problems:  For a body with mass density ρ (mass per unit volume) we have that

  • The total mass is  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev
  • The position of the center of mass is  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev
  • The mass moment of inertia about the center of mass is

  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

 where d = r − rG

For 2D problems: We know the COM must lie in the i,j plane and we don’t need to calculate the whole matrix.  

For a body with mass per unit area µ we can therefore use the formulas

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

  • The total mass is  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev
  • The position of the center of mass is   Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev
  • The mass moment of inertia about the center of mass is  Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

where d = r − rG

Example 1: To show how to use these, let’s calculate the total mass, center of mass, and mass moment of inertia of a rectangular prism with faces perpendicular to the i, j,k axes:

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

First the total mass (sort of trivial)

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Now the COM

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

And finally the mass moment of inertia

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Example 2: As a second example, let’s calculate the mass moment of inertia of a cylinder with mass density ρ , length L and radius a.  (We know the COM is at the center and we know the total mass so we won’t bother calculating those). We have to do the integral with polar coordinates

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Now note that d =r cosθ dy = r sinθ dz = z .  We can have Mupad do the dirty work:

[reset() :

[ds := r*cos(q): dy := r*sin(q): dz := z:

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev  

Example 3: Let’s finish up with a 2D example. Find the mass, center of mass, and out of plane mass moment of inertia of the triangle shown in the figure.

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

The total mass is   Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

The position of the COM is 

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

The 2D mass moment of inertia is

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

This is all a big pain, and you may be contemplating a life of crime instead of an engineering career.   Fortunately, it is very rare to have to do these sorts of integrals in practice, because all the integrals for common shapes have already been done. You can google most of them. The tables below give a short list of all the objects we will encounter in this course.

Table of mass moment of inertia tensors for selected 3D objects

Prism

M = ρ abc

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Solid Cylinder

M = πρ a2L

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Solid Cone

M =π/h ρα2h

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Solid Sphere

M = 4/3 πρα3

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Solid Ellipsoid

M =4/3 πραbc

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Hollow Cylinder

M = πρ (b2 − a2)L

Linear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev

Table of mass moment of inertia about perpendicular axis for selected 2D objects

SquareLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevIGzz = M/12 (α2 + b2)
DiskLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevIGzz = M/2R2
Thin ringLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevIGzz = -MR2
Hollow diskLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevIGzz = M/2 (α2 + b2)
Slender rodLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRevIGzz =  M/12 L2
Triangular PlateLinear Momentum, Angular Momentum and Kinetic Energy of Rigid Bodies (Part - 2) Civil Engineering (CE) Notes | EduRev M/18 (α2 + b2)

 

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