Essential Physics 1 F. Firk

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energy function can be found that depends only on the position of the object (or objects).
4.5.2 Conservative forces
Let Fx and Fy be the Cartesian components of the forces acting on a moving particle
with coordinates [x, y]. The work done W1�!2 by the forces while the particle moves from
the position P1[x1, y1] to another position P2[x2, y2] is
W1�!2 = +"[x1, x2] Fxdx + +"[y1, y2] Fydy (4.28)
= +"[P1, P2] (Fxdx + Fydy) .
If the quantity Fxdx + Fydy is a perfect differential then a function U = f(x, y) exists
such that
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Fx = "U/"x and Fy = "U/"y . (4.29)
Now, the total differential of the function U is
dU = ("U/"x)dx + ("U/"y)dy (4.30)
= Fxdx + Fydy.
In this case, we can write
+" = +" dx + Fydy) = U = f(x, y).
dU (F
x
The definite integral evaluated between P1[x1, y1] and P2[x2, y2] is
+"[P1, P2] (Fxdx + Fydy) =f(x2, y2) f(x1, y1) = U2 U1 . (4.31)
We see that in evaluating the work done by the forces during the motion, no mention is
made of the actual path taken by the particle. If the forces are such that the function
U(x, y) exists, then they are said to be conservative. The function U(x, y) is called the
force function.
The above method of analysis can be applied to a system of many particles, n. The
total work done by the resultant forces acting on the system in moving the particles from
their initial configuration, i, to their final configuration, f, is
Wi�!f = "[k=1, n] +"[Pk1, Pk2] (Fkxdxk + Fkydyk), (4.32)
= Uf Ui,
a scalar quantity that is independent of the paths taken by the individual particles.
Pk1[xk1, yk1] and Pk2[xk2, yk2] are the initial and final coordinates of the kth-particle.
The potential energy, V, of the system moving under the influence of conservative
forces is defined in terms of the function U: V a" U .
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Examples of interactions that take place via conservative forces are:
1) gravitational interactions
2) electromagnetic interactions
and
3) interactions between particles of a system that, for every pair of particles, act
along the line joining their centers, and that depend in some way on their distance apart.
These are the so-called central interactions.
Frictional forces are examples of non-conservative forces.
There are two other major methods of solving dynamical problems that differ in
fundamental ways from the method of Newtonian dynamics; they are Lagrangian dynamics
and Hamiltonian dynamics. We shall delay a discussion of these more general methods
until our study of the Calculus of Variations in Chapter 9.
4.6 Particle interactions
4.6.1 Elastic collisions
Studies of the collisions bewteen objects, first made in the 17th-century, led to the
discovery of two basic laws of Nature: the conservation of linear momentum, and the
conservation of kinetic energy associated with a special class of collisions called elastic
collisions.
The conservation of linear momentum in an isolated system forms the basis for a
quantitative discussion of all problems that involve the interactions between particles.
The present discussion will be limited to an analysis of the elastic collision between two
particles. A typical two-body collision, in which an object of mass m1 and momentum p1
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makes a grazing collision with another object of mass m2 and momentum p2 (p2
shown in the following diagram. (The coordinates are chosen so that the vectors p1 and p2
have the same directions). After the collision, the two objects move in directions
characterized by the angles � and � with momenta p1� and p2�.
Before After
m1 p1�
�
m1 p1 m2
p2 �
m2 p2�
If there are no external forces acting on the particles so that the changes in their
states of motion come about as a result of their mutual interactions alone, the total linear
momentum of the system is conserved. We therefore have
p1 + p2 = p1� + p2� (4.33)
or, rearranging to give the momentum transfer,
p1 p1� = p2� p2 .
The kinetic energy of a particle, T is related to the square of its momentum
(T = p2/2m); we therefore form the scalar product of the vector equation for the
momentum transfer, to obtain
p12 2p1�"p1� + p1�2 = p2�2 2p2��"p2 + p22 . (4.34)
Introducing the scattering angles � and �, we have
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p12 2p1p1�cos� + p1�2 = p22 2p2p2�cos� + p2�2 .
This equation can be written
p1�2 (x2 2xcos� + 1) = p2�2(y2 2ycos� + 1) (4.35)
where
x = p1/p1� and y = p2/p2� .
If we choose a frame in which p2 = 0 then y = 0 and we have
x2 2xcos� + 1 = (p2�/p1�)2 . (4.36)
If the collision is elastic, the kinetic energy of the system is conserved, so that
T1 + 0 = T1� + T2� (T2 = 0 because p2 = 0) . (4.37)
Substituting Ti = pi2/2mi , and rearranging, gives
(p2�/p1�)2 = (m2/m1)(x2 1) .
We therefore obtain a quadratic equation in x:
x2 + 2x(m1/(m2 m1))cos� [(m2 + m1)/(m2 m1)] = 0 .
The valid solution of this equation is
x = (T1/T1�)1/2 = (m1/(m2 m1))cos�
+ {(m1/(m2 m1))2cos2� + [(m2 + m1)/(m2 m1)]}1/2. (4.38)
If m1 = m2, the solution is x = 1/cos�, in which case
T1�= T1cos2� . (4.39)
In the frame in which p2 = 0, a geometrical analysis of the two-body collision is
useful. We have
p1 + ( p1�) = p2�, (4.40)
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leading to
p1�
p1 �
� �
p2� p1�
If the masses are equal then
p1� = p1cos� .
In this case, the two particles always emerge from the elastic collision at right angles to
each other (� + � = 90o).
In the early 1930 s, the measured angle between two outgoing high-speed nuclear
particles of equal mass was shown to differ from 90o. Such experiments clearly
demonstrated the breakdown of Newtonian dynamics in these interactions.
4.6.2 Inelastic collisions
Collisions between everyday objects are never perfectly elastic. An object that has
an internal structure can undergo inelastic collisions involving changes in its structure.
Inelastic collisions are found to obey two laws; they are
1) the conservation of linear momentum
and
2) an empirical law, due to Newton, that states that the relative velocity of the
colliding objects, measured along their line of centers immediately after impact, is
e times their relative velocity before impact.
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