Euler’s Second Law -- Motion in xy Plane

Euler’s second law requires that the sum of the moments of the external forces and external couples acting on the rigid body equals to the change in the angular momentum of the rigid body ** .

Form 2: Change of angular momentum about its center of mass for a rigid body

Σ M_C = I_ZZ^C α

Euler’s Second Law – Mass Center Form

where Σ M_C is the sum of all external moments acting on the rigid body about its mass center

I_ZZ^C is the mass moment of inertia of the rigid body about its mass center, C

α is the angular acceleration of the rigid body; α = α k

Form 3: Change of angular momentum about any point P, in the body or not.

Σ M_P = I_ZZ^P α + ( r_PC x m a_P )_z

Σ M_P = I_ZZ^C α + ( r_PC x m a_C )_z

Euler’s Second Law – Forms about Point P

where Σ M_P is the sum of all external moments acting on the rigid body about point P

I_ZZ^P is the mass moment of inertia of the rigid body about any point P

I_ZZ^C is the mass moment of inertia of the rigid body about its mass center, C

α is the angular acceleration of the rigid body

m is the mass of the rigid body

r_PC is the position vector from P to C, where P is any point and C is at

the mass center of the rigid body; ( )_z – refers to z-component only

Strategy:

Step 1: Identify all moments of the external forces and external couples acting on the rigid body

from your free body diagram. i.e. Σ M_C or Σ M_P

Step 2: Apply Euler’s second law (equations of motion). In plane motion, there is one scalar equation

of motion governing the rotational motion of the rigid bod

Σ M_C = I_ZZ^C αk or Σ M_P = I_ZZ^P α + ( r_PC x m a_P )_z or Σ M_P = I_ZZ^C α + ( r_PC x m a_C )_z

Step 3: Couple this equation with the two scalar equations of motion from Euler’s first Law

i.e. Σ F_x = m a_Cx Σ F_y = m a_Cy (or polar or intrinsic forms)

and determine the unknown quantities in the set of three equations. You may need more information

to relate the acceleration of the center of mass to angular acceleration using kinematics.

Click here for examples. ** Restriction: Rigid body is symmetric about its z-axis.

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