Angular Impulse-Momentum Principle

 

System of rigid bodies

 

The total angular momentum is equal to the sum of the momenta of the individual bodies.

 

Pure translation

 

Translation with respect to a fixed-point O.

 

∫∑M_O dt = ∑(r_G/Oi,2 x m_iv_i,2) - ∑(r_G/Oi,1 x m_iv_i,1)

 

Pure rotation

 

Rotation about fixed-axis through point O.

 

∫∑M_O = ∑(I_Oi,2ω_i,2) - ∑(I_Oi,1ω_i,1)

 

General planar motion

 

General planar motion with respect to the body's mass center.

 

∫∑M_G = ∑(I_Gi,2ω_i,2) - ∑(I_Gi,1ω_i,1)

 

General planar motion with respect to a fixed-axis through point O.

 

∫∑M_O = ∑(r_G/Oi,2 x m_iv_Gi,2 + I_Gi,2ω_i,2) - ∑(r_G/Oi,1 x m_iv_Gi,1 + I_Gi,1ω_i,1)

 

Conservation of angular momentum

 

Angular momentum is conserved if the externally applied moments sum to zero.

 

∑(I_Gi,2ω_i,2) = ∑(I_Gi,1ω_i,1)

 

∑(r_G/Oi,2 x m_iv_Gi,2 + I_Gi,2ω_i,2) = ∑(r_G/Oi,1 x m_iv_Gi,1 + I_Gi,1ω_i,1)