Kinetics – Body Translating in a Plane

Key Concepts: Euler's First Law governs the translation of a rigid body in plane motion.

Translation implies there is no rotational acceleration of the rigid body. You may be asked to

find the acceleration of the center of mass given the external forces or vice-versa find the

forces given the acceleration of the center of mass.

In a Nut Shell: A body translating in a plane responds to the forces acting on the body. The

strategy to solve for the motion of a rigid body translating in a plane involves three key steps.

Step 1: Draw a free body diagram showing all external forces acting on the rigid body.

Step 2: Write down Euler’s 1^st law (equations of motion) for the body. In plane motion,

there are two scalar equations of motion governing the motion of its center of mass. In

vector form this law is:

ΣF = m a_c = m dv_c/dt = m a_c (vector equation)

where ΣF are the forces acting on the body, m is its mass, and a_c is acceleration of its

mass center

Step 3: Check to see the number of unknowns in the scalar equations of motion equal the

Number of equations. These unknowns typically include forces and accelerations. Frequently

there are more unknowns than scalar equations. So you will need to supplement the scalar

equations of motion with other relations. i.e for sliding or impending sliding, friction = μ N

Once determined, the acceleration of the mass center can be integrated to find speed and position.

In scalar form, Euler’s 1^st law takes the following forms:

Coordinate System	Component	Component
Rectangular	Σ Fx = macx	Σ Fy = macy
Intrinsic	Σ Ft = mact	Σ Fn = macn
Polar	Σ Fr = macr	Σ Fθ = macθ

and for translation

Σ M_c = 0

where ΣM_c is the moment of the external forces about C, the center of mass of the body

Click here for examples.

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