Engr Help

Principle of Angular Impulse and Momentum in a Plane

Key Concept: The principle of angular impulse and momentum is just an integrated form of

Euler's Second Law by integrating with respect to time. Any analysis can start directly with

Euler's Second Law or with the application of the principle of angular impulse and momentum.

In a Nut Shell: The principle of angular impulse and momentum in a plane is that the

angular impulse acting on a body equals its change in angular momentum during the

time of the impulse, say from t₁ to t₂.

Σ M_C = I^C_zz α

Euler’s Second Law

Integration with respect to time yields (Note rotational acceleration about z-axis.)

t₂

∫ Σ M_C k dt = I^C_zz ω₂ k - I^C_zz ω₁ k

t₁

Principle of Angular Impulse and Momentum

where

t₂

∫ Σ M_C k dt = the angular impulse acting on the body from t₁ to t₂

t₁

Σ M_C = sum of moments of all external forces about the center of mass and all

external couples

I^C_zz = mass moment of inertia of body about its mass center, C

ω₂ = angular speed at time t₂ , ω₁ = angular speed at time t₁

Strategy:

Step 1: Draw a free body diagram showing all external forces and couples acting on

the body. Calculate the moment of all external forces and couples on the body that

result in a change in its angular momentum.

Step 2: Calculate the angular impulse and set it equal to the change in angular momentum

about the center of mass, C