University of California at Berkeley
J. Lubliner
CE C30/ME C85
Spring 2008
Handout 1
Statics of particle systems
A particle is in equilibrium if the resultant force on it is zero. Now, consider a discrete system of
N particles. Let the resultant force on the i-th particle be denoted Fi . It can be thought of as the
vector sum of the forces exerted on this particle by each of the other particles in the system, Fij ,
j 6= i, and the external force Fie , that is,
Fi = Fie +
X
Fij .
j6=i
If Fi = 0 then, of course,
PN
i=1 Fi
= 0. Now,
N
X
Fi =
i=1
N
X
Fie +
i=1
N X
X
Fij .
i=1 j6=i
But the double sum can be written as
N X
X
Fij =
i=1 j6=i
N X
i−1
X
N X
N
X
Fij +
i=1 j=1
Fij
i=1 j=i+1
and the second double sum on the right can be rewritten, by interchanging the indices i and j, as
N X
N
X
Fji or, equivalently, as
j=1 i=j+1
N X
i−1
X
Fji , since either way it represents the summation over all
i=1 j=1
particle pairs i, j such that i > j. Consequently,
N X
X
Fij =
i−1
N X
X
(Fij + Fji )
i=1 j=1
i=1 j6=i
According to Newton’s third law of motion, the forces between the particles are equal and
opposite, that is, Fij = −Fji . It follows that
N X
X
Fij = 0, and therefore
N
X
i=1
i=1 j6=i
Fi =
N
X
Fie .
i=1
Finally, then, a system of particles is in equilibrium only if
N
X
Fie = 0,
(1)
i=1
that is, the vector sum of the external forces on the system is zero.
Another property of a force between two particles is that, if it is not magnetic, it is directed
along the line joining the particles. Now, if the position vector of particle i (with respect to an
arbitrary origin O) is ri , then the vector from particle i to particle j is rj − ri , and since the
cross-product of two parallel vectors is zero, it follows that (rj − ri ) × Fij = 0. Now
N
X
i=1
ri × Fi =
N
X
i=1
ri × Fie +
N X
X
i=1 j6=i
ri × Fij
i−1
and the double sum, with a similar manipulation as before, can be rewritten as N
i=1
j=1 (ri −
rj ) × Fij = 0. Consequently, another necessary condition for the equilibrium of a particle system is
P
N
X
ri × Fie = 0,
P
(2)
i=1
that is, the vector sum of the moments of the external forces on the system is zero.
A particle system is rigid if the distances between all pair of particles remain unchanged. Since
the equilibrium equations (1) and (2) are independent of the number and nature of the particles,
they are also the equilibrium equations for rigid bodies in general if we write simply Fi in place of
Fie , where the index i denotes the point of application of the force.
Problems
H1-1. Three particles are located on the vertices of an equilateral triangle. If F1e is of known
magnitude F and is directed along the line from particle 1 to particle 2, while F2e is of unknown
magnitude and its direction is perpendicular to that line, find F3e (showing its direction on a sketch)
so that equilibrium is satisfied.
H1-2. If in the preceding problem the force F2e were directed along the line from particle 2 to
particle 3, would the problem have a solution? Explain your answer.