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Unformatted text preview: he particle, i. e. © E. F. Schubert 1 p(t ) = m v(t ) = m dx (t )
dt = const. (1.1) Newton’s second law relates an external force, F, to the second derivative of the position x (t )
with respect to t, =m F d 2 x(t )
dt2 = ma (1.2) where a is the acceleration of the particle. Newton’s first and second law provide the state
variables x(t) and p(t) in the presence of an external force.
Newton’s second law is the basis for the introduction of work and energy. Work done by moving
a particle along the x axis from 0 to x by means of the force F(x) is defined as W ( x) = x ∫0 F ( x) dx . (1.3) The energy of the particle increases by the (positive) value of the integral given in Eq. (1.3). The
total particle energy, E, can be (i) purely potential, (ii) purely kinetic, or (iii) a sum of potential
and kinetic energy. If the total energy of the particle is a purely potential energy, U(x), then
W(x) = − U(x) and one obtains from Eq. (1.3) F ( x) = − d
U ( x) .
dx (1.4) If, on the other hand, the total energy is purely...
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- Fall '12