Dynamics of a system of particles

Which one is correct about the first moment of mass w.r.t. center of mass c.?

• Σr̰_i × m_i = 0

• Σr̰_i × m_i ≠ 0

• Σρ_i × m_i = 0

• Σρ_i × m_i ≠ 0

The center of mass r̰_c is

• ∑(r̰_i × m_i)/m

• (∑r̰_i × m)/∑m_i

• ∑(r̰_i² × m_i)/m

• ∑(r̰_i² × m_i)/∑m_i

Linear momentum P̰ of the system is

• P̰ = Σ(m_i × v̰_i)

• P̰ = m × v̰_c

• P̰ = m × d(v̰_c)/dt

• P̰ = Σ(m_i × d(v̰_i)/dt)

An angular momentum H̰_o of the system about O is defined as

• Σ(r̰_i ×m_iv̰_i)

• Σ(ρ_i ×m_i d(ρ_i)/dt)

• Σ(r̰_i ×m_i d(ρ_i)/dt)

• Σ(ρ_i ×m_i v̰_i)

An angular momentum H̰_c of the system about the center of mass c is defined as

• Σ(r̰_i ×m_i v̰_i)

• Σ(ρ_i ×m_i d(ρ_i)/dt)

• Σ(r̰_i ×m_i d(ρ_i)/dt)

• Σ(ρ_i ×m_i∙v̰_i)

What is the relation of the angular momentums of the system about O and c (H̰_o and H̰_c)?

• H̰_c = H̰_o+(r̰_c × mv̰_c)

• H̰_o = H̰_c+(r̰_c × mv̰_c)

• H̰_c = H̰_o+(dr̰_c/dt × mv̰_c)

• H̰_o = H̰_c+(dr̰_c/dt × mv̰_c)

The Law of linear momentum states that

• Rate of change of the linear momentum of the system is equal to the force sum applied to all particles.

• Rate of change of linear momentum of the system is proportional to the equivalent acceleration at c.

• Sum of all internal forces in the system is zero.

• Sum of the applied forces to all particles is proportional to the linear momentum of the system.

The Law of angular momentum states that

• Rate of change of the angular momentum of the system about O is equal to the moment sum about O of all applied forces to the particles.

• Rate of change of the angular momentum of the system about c is equal to the moment sum about c of all applied forces to the particles.

• Both statements are correct.

• Both statements are incorrect.

Which one is the expression of the law of linear momentum?

• P̰ = mv̰_c

• dP̰/dt = ΣF̰_a

• ΣF̰_a = m dv̰_c /dt

• No correct answers

Which formula is correct:

• M̰_o = d(H̰_o)/dt

• M̰_c = d(H̰_c)/dt

• H̰c = Σ(ρ_i×m_i dρ_i/dt)

• All are correct.