Integrated forms of Euler’s Second Law

In a Nut Shell: Euler’s second law deals with change in angular momentum of a rigid body. One option is to use this law directly (as given below). Another option is to use integrated forms of this law.

M_C = dH_C/dt

Euler’s Second Law

where M_C is the sum of all external moments acting on the rigid body about its mass center

dH_C/dt is the rate of change of the angular momentum of the rigid body about its

center of mass measured in an inertial frame of reference dH_C/ dt = I_C dω/dt

If the external moments are a function of time, then integration of Euler’s second law with respect

time yields the principle of angular impulse and momentum. (See table below.)

Start with dH_C/ dt = I_C dω/dt . Then M_C dt = I_Cdω and integration from time 1 to time 2 gives

2 2

∫ M_C(t) dt = ∫ I_C dω = I_C ω₂ - I_C ω₁

1 1

Angular Impulse = Change in Angular Momentum

Click here for an example of the principle of angular impulse and momentum.

If the moments are a function of angular position, then integration with respect to angular displacement yields the principle of work and energy for angular motion. (See table below.)

Start with M_C = I_C dω/dt, take the dot product with the displacement, dθ k, and then integrate

from position 1 to position 2.

M_C ∙ dθ k = I_C dω/dt ∙ dθ k = I_{C [}dω/dt ∙ dθ/dt]dt k = I_C [dω/dt ∙ ω]dt = I_C dω∙ ω = I_C ω∙ dω

2 2

∫ M_C ∙ dθ = ∫ I_C ω∙ dω = ½ I_Cω₂² – ½ I_C ω₁² = T₂ – T₁

1 1

Work done by moments = Change in Kinetic Energy

Click here for an example of Work & Energy.

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