Planar Kinetics: Force & Acceleration - Dynamics

Mass Moment of Inertia

F =ma studies the translational motion
M = Iα studies the rotational motion

• M is the moment
• I is the mass moment of inertia. The moment of inertia is the measure of resistance to angular acceleration(M = Iα), similar to mass is a measure of resistance to acceleration(F = ma)
• α is the angular acceleration

An object (such as a flywheel) with a large moment of inertia will be difficult to stop once it has been set into motion.

The integral of the second moment is the moment of inertia: I = ∑ ∫ r^2 dm The moment arm, r, is the perpendicular distance from the axis to the element dm. The units used for measurement are Kg*m^2 or slug*ft^2

Parallel-Axis Theorem: If the moment of inertia passing through the center of mass for the object is know then any other parallel moment of inertia can be found.
I = ∑(IG + md^2)

• IG is the moment of inertia that passes through G(center of mass)
• m is the mass of the body
• d is the perpendicular distance between the two parallel axes

Radius of Gyration(k): A geometrical property with units of length. This is related to the moment of inertia by: I = mk^2


Equations of Motion for Translation

All particle on a body will move with the same acceleration. Since there is no rotation α = 0.
For Rectilinear Translation(all particles traveling in straight parallel paths)

• ∑Fx = m(aG)x
• ∑Fy = m(aG)y
• ∑MG = 0 or to sum moments about another point, P, ∑MP = (maG) d where d is the perpendicular distance from the line of action of maG to the point, P.

For Curvilinear Translation(all particles travel curved paths that are parallel to one another)

• ∑Fn = m(aG)n
• ∑Ft = m(aG)t
• ∑MG = 0

Equations of Motion for Rotation about a Fixed Axis

When a body moves about a fixed axis, the center of mass, G will move in a circular path.

• ∑Fn = m(aG)n also equals = m ω^2 rG
  
• ∑Ft = m(aG)t also equals = m α rG
  
• ∑MG = IG α

Equations of Motion for General Plane Motion

When a body translates and rotates

• ∑Fx = m(aG)x
• ∑Fy = m(aG)y
• ∑MG = IG α

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