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Integrated forms of Euler’s First Law

In a Nut Shell: Euler’s first law deals with change in linear momentum of a particle or of the

center of mass of a rigid body. One option is to use this law directly (as given below). Another

option is to use integrated forms of this law.

F = m dv/dt

Euler’s First Law

NOTE: Euler’s first law is a vector equation where

F are all the external forces , m is the mass of the particle (or entire body)

dv/dt is the acceleration of the center of mass in the inertial frame of reference

If the external forces are a function of time, then integration of Euler’s first law with respect

time yields the principle of linear impulse and momentum. (See table below.)

Start with F = mdv/dt . Then F dt = mdv and integration from time 1 to time 2 gives

2 2

∫ F(t) dt = ∫ mdv = mv₂ - mv₁

1 1

Linear Impulse = Change in Linear Momentum

Note: The principle of linear impulse and momentum is a vector equation. Click for example.

If the forces are a function of position, then integration with respect to displacement yields the

principle of work and energy. (See table below.)

Start with F = mdv/dt , take the dot product with the displacement, dr, and then integrate

from position 1 to position 2.

F ∙ dr = mdv /dt ∙ dr = mdv /dt ∙ (dr /dt) dt = mdv /dt ∙ v dt) = m dv ∙ v = m v ∙ dv

2 2

∫ F ∙ dr = ∫ m v ∙ dv = ½ m v₂² – ½ m v₁² = T₂ – T₁

1 1

Work done = Change in Kinetic Energy

Click here for an example of Work & Energy.