University Physics I - Classical Mechanics, 2019

7.6. IN SUMMARY 151
which is equal to 1 J/s. The average power going into or coming out of a system by mechanical
means, that is to say, through the action of a force applied at a point undergoing a displacement
Δx, will be
P av = ΔE
Δt = W Δt = F Δx
(7.29)
Δt
assuming the force is constant. Note that in the limit when Δt goes to zero, this gives us the
instantaneous power associated with the force F as
P = Fv (7.30)
where v is the (instantaneous) velocity of the point of application of the force. This one-dimensional
result generalizes to three dimensions as
using again the dot product notation.
P = ⃗ F · ⃗v (7.31)
An important goal of this chapter has been to develop a set of tools that you may use to find
out where power is spent, and how much: in any practical situation, which systems are giving
energy and which are taking it in, what forms of energy conversion are taking place, and where and
through which means are the exchanges and conversions happening. These are extremely important
practical questions; the problems and exercises that you will see here will give you a feel for the
variety of situations that can already be analyzed by this “systems-based” approach, but in a way
they will still do little more than scratch the surface.
7.6 In summary
1. The change in the momentum of a system produced by a force ⃗ F acting over a time Δt is
given the name of “impulse” and denoted by ⃗ J. For a constant force, we have ⃗ J =Δ⃗p = ⃗ F Δt.
2. Work, or “doing work” is the name given in physics to the process by which an applied force
brings about a change in the energy of an object, or of a system that contains the object on
which the force is acting.
3. The work done by a constant force ⃗ F acting on an object or system is given by W = ⃗ F · Δ⃗r,
where the dot represents the “dot” or “scalar” product of the two vectors, and Δ⃗r is the
displacement undergone by the point of application of the force while the force is acting. If
the force is perpendicular to the displacement, it does no work.
4. For a system that is otherwise closed, the net sum of the amounts of work done by all the
external forces is equal to the change in the system’s total energy, when all the types of energy
are included. Note that, for deformable systems, the displacement of the point of application
may be different for different forces.