Impulse and Momentum of a Rigid Body

Impulse and momentum provides a relationship between velocity and force. An impulse is a force integrated over time, while momentum is a product of mass and velocity.

Momentum

For a rigid body there are two types of momentums that could be applied. These are linear momentum, and angular momentum. Linear momentum is based off of the rigid body’s velocity and the rigid body’s mass, which can be seen in equation 1. Angular momentum however, is based off of the rigid body’s angular velocity and its mass moment of inertia. Refer to equation 2 to view how to calculate angular momentum.

(Eq 1) $L=mv$

m = mass

v = velocity

L = Linear Momentum

(Eq 2) $H=Iϖ$

I = Mass Moment of Inertia

ϖ = Angular Velocity

H = Rotational Momentum

Impulse

There are also angular and linear impulses that can be placed on a rigid body. A linear impulse will be a force integrated over time, which can be seen in equation 3. An angular impulse on the other hand would be a moment integrated over time as seen in equation 4.

(Eq 3)  $Σ∫_{t_1}^{t_2}Fdt = Σ(Ft_2-Ft_1)$

F = Force

(Eq 4) $Σ∫_{t_1}^{t_2}Mdt=∑(Mt_2-Mt_1)$

M = Momentum

Impulse and Momentum x-y Plane

The relationship between impulse and momentum for a rigid body on the x-y plane can be seen in equations 5-7.

(Eq 5) $mv_{x_1}+∑∫_{t1}^{t2}Fdt=mv_{x_2}$

(Eq 6) $mv_{y_1}+∑∫_{t1}^{t2}Fdt=mv_{y2}$

(Eq 7) $Iϖ_1+∑∫_{t1}^{t2}Mdt=Iϖ_2$

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