Page 179 - Standard Handbook Of Petroleum & Natural Gas Engineering

164 General Engineering and Science

From Tables 2-6 and 2-7 and the parallel axis theorem,

2g sin 30'
a= = 3.57rps2
3r
ax = ra = 10.73 ft/s2

Conservation of Momentum. If the mass of a body or system of bodies remains
constant, then Newton's second law can be interpreted as a balance between force
and the time rate of change of momentum, momentum being a vector quantity defined
as the product of the velocity of a body and its mass.

d d
F = ma = -mv = -G (2-30)
dt dt
Integrating Equation 2-30 with respect to time yields the impulse/momentum
equation

JFdt = AG (2-31)

where Fdt is called the impulse, and AG is the change in momentum. Equation 2-31
can be applied explicitly and is particularly useful when the force is known as a function
of time.
In collisions between two bodies the contact force and the duration of contact
are usually unknown. However, the duration of contact is the same for both bodies,
and the force on the first body is the negative of the force on the second body.
Thus the net change in momentum is zero. This is called the principle of conservation
of momentum.
If a collision is purely plastic, then the two colliding bodies will adhere to each
other and move on as a single body. Knowing the initial velocities and masses thus
allows calculation of the final velocity.

mlvl + m2v2 = (m, + m,)v (2-32)
If the collision is purely elastic or elasto-plastic, then the two bodies will depart the
collision with different velocities.

m,v,, + m2v21 = m,v,2 + m2v22 (2-33)
In this case, an additional equation is required before the final velocities may be
found. Thus, the coefficient of restitution e is defined as the ratio of the velocity of
separation to the velocity of approach:

(2-34)