Impulse and Momentum for Systems of Particles

 

Linear momentum

 

The total linear momentum of a system of particles is the sum of the linear momenta of the individual particles making up that system. The linear momentum may also be calculated by using the system's mass and the system's mass center velocity.

 

G = ∑m_iv_i = mv_G

 

Linear impulse

 

The total linear impulse is equal to the impulse applied by an equivalent resultant force ∑F applied to the system

 

I = ∫∑F dt

 

Linear impulse-momentum principle

 

The total linear impulse equals the change in linear momentum of the system.

 

∫∑F dt = ∑m_iv_i,2 - ∑m_iv_i,1

 

Angular momentum

 

The total angular momentum of a system of particles is equal sum of the angular momenta of the individual particles. You can also use the position, velocity and mass moment of inertia of the system’s mass center.

 

H_O = ∑(r_i x m_iv_i) = r_G x mv_G + I_Gω

 

Angular impulse

 

The total angular impulse is equal to the sum of the individual force impulses. Note that the forces may act at different locations.

 

J_O = ∫∑(r_i x F_i ) dt

 

Angular impulse-momentum principle

 

The total angular impulse applied to the system is equal to the change in angular momentum.

 

∫∑(r_i x F_i ) dt = ∑(r_i,2 x m_iv_i,2) - ∑(r_i,1 x m_iv_i,1)