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In this section, we collect together all the important formulas from the preceding sections, and summarize the equations that we use to analyze motion of a rigid body.

We consider motion of a rigid body that has mass density ρ during some time interval t_{0} < t < t_{1} , and define the following quantities:

**6.6.1 Forces, torques, impulse, work, power**

- The total force acting on the body
- The total linear impulse exerted by forces during the time interval
- The total moment (including torques) acting on the body
- The tot al angular impulse exerted on the body during the time interval
- The rate of work done by forces and torques acting on the body
- The total work done by forces and torques on the body during the time interval

**6.6.2 Inertial properties**

- The total mass is
- The position of the center of mass is
- The mass moment of inertia about the center of mass

where d = r − r_{G}

For a 2D body with mass per unit area µ we use

- The total mass is
- The position of the center of mass is
- The mass moment of inertia about the center of mass is

where d = r − r_{G }

**6.6.3 Describing motion**

- The rotation tensor (matrix) maps the vector connecting two points in a solid before it moves to its position after motion r
_{B}− r_{A}= R(p_{B}− p_{A})

- The spin tensor is related to R by
- Rotation through an angle θ about an axis parallel to a unit vector n = n
_{x}i +n_{y}j + n_{z}k is

- The angular velocity vector ω = ω
_{ x }i +ω_{y}j+ ω_{ z}k is related to W by

- The angular acceleration vector is α = dω/dt
- The velocities of two points A and B in a rotating rigid body are related by v
_{B}−v_{A}= ω ×(r_{B}− r_{A}) - The accelerations of A and B are related by a
_{B}− a_{A}= α ×(r_{B}−r_{A}) + ω×(v_{B}− v_{A}) = α ×(r_{B}−r_{A}) + ω× ω×(r_{B}−r_{A})]

**6.6.4 Momentum and Energy **

- The total linear momentum is p = MvG
- The angular momentum (about the origin) is h =rG × MvG+ IG ω
- The total kinetic energy is T= 1/2 Mv
_{G}·v_{G}+1/2 ω·I_{G}ω

For 2D planar problems, we know ω = ω_{z}k . In this case, we can use

- The total linear momentum is p = MvG
- The total angular momentum (about the origin) is h =r
_{G}× Mv_{G}+IG_{zz}ω_{z}k - The total kinetic energy is

**6.6.5 Conservation laws **

- Linear momentum
- Angular momentum
- Work-Power - Kinetic Energy relation
- Energy equation for a conservative system d/dt (T+V) =0 T
_{0}+ V_{0}= T_{1}+V_{1}

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