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Conservation of Momentum

Linear Conservation of Momentum

When there is a system of rigid bodies that interact with each other, the linear momentum of the objects that interact with each other will be conserved if all of the impulses cause by forces sum to zero.  This is known as linear conservation of momentum.  As a result, the linear momentum of the system will be constant.

(Eq 1)  $m_1v_1=m_2v_2$

$m$ = mass

$v$ = velocity

Notice from the above equation, even though the linear momentum is constant, the velocity may not be.  If on one side of the equation the mass is greater than on the other side of the equation the velocity will differ.  As a result the side that has the least mass will have a higher velocity than the side that has more mass.  This is because linear momentum is a product of both mass and velocity.

Angular Conservation of Momentum

Unlike a particle, a rigid body has the capability to rotate about a fixed axis.  As a result there will an angular momentum that is a product of the bodies angular velocity and its mass moment of inertia.  However, the principles are still the same.  If the impulses on a system of rigid bodies caused by moments acting on the rigid body sums to zero the angular momentum will be conserved.  Due to this fact the angular momentum of the system will be constant.

(Eq 2)  $I_1ϖ_1=I_2ϖ_2$

$I$ = Moment of Inertia

$ϖ$ = anglular velocity

Just like with linear momentum, for angular moment, even if the momentum is constant the angular velocity may not be.  If the mass moment of inertia differs on each side of the equation than the angular velocity will also differ.  In turn this will cause the side that has the higher mass moment of inertia to have a lower angular velocity.  In contrast the side with lower mass moment of inertia will have a higher angular velocity.

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