Page 181 - Standard Handbook Of Petroleum & Natural Gas Engineering

166 General Engineering and Science

where V is the velocity of the center of mass and d is the perpendicular distance
between the vector v and the point 0. And if 0 is a fixed axis or instantaneous center
of zero velocity, then

H, = I,o (2-37)
Likewise, the angular impulse is defined as

I M,dt = AH,, (2-38)

In collisions, angular momentum, like linear momentum, is conserved.

Conservation of Energy
In a rigid-body system, energy is conserved in the sense that the net change in
mechanical energy must be equal to the net work done on the system.

U = AT + AVg + AVe (2-39)
U is the net work done on the system and is defined as the sum of the work done
by external forces and external moments.

U =jF*ds+jM*dB (2-40)

The work of the force F is positive if it acts in the direction of the displacement ds,
and the work of the moment M is positive if it acts in the direction of rotation de.
AT is the change in kinetic energy, made up of a change in linear kinetic energy
and rotational kinetic energy.

1 1-
AT = -m(C:-Vy)+-I(mt-of)
2 2
If the body in question has a fixed axis or an instantaneous center of zero velocity,
then Equation 2-41 can be simplified to

AVg is the net change in gravitational potential energy. This term is path
independent and depends only on the initial and final heights, h, and h, above some
arbitrary reference height with respect to the surface of the earth.
AVg = mg(h, - hi) = W(h, - h,) (2-43)
AVe is the net change in elastic energy stored in a massless spring, due to extension
or compression (no spring is massless, but this assumption is reasonably accurate for
most engineering calculations).
1
AVe = - k(X; - Xy) (2-44)
2