The University of Alabama at Birmingham (UAB)
Department of Physics
PH 461/561 – Classical Mechanics I – Fall 2005
Assignment # 5 Due: Thursday, September 15
(Turn in for credit!)
Activities based on previous lectures:
1. A particle of mass m moves in one dimension under the force:
(
)
2
() 2 e e
ax ax
Fx aA −−
=− where a > 0, A > 0
a) Discuss the effect of this force on the total mechanical energy of the particle.
b) Find an expression for the potential energy V(x) of the particle.
c) Graph this potential energy by hand (Computer use for checking ok)
d) Provide a qualitative discussion of the possible types of motion depending on the total
energy E. (bound, unbound motion, turning points, etc.)
e) Find the position of the point of equilibrium.
f) For total energy slightly above –A, the motion of the particle is periodic and the potential
energy may be approximated by parabolic well. In order to prove this perform the
following steps
f-1) Expand V (x) in a power series around x = 0 keeping only terms up to second order.
f-2) Calculate the period of small oscillations around the point of equilibrium
(i.e., the minimum of the potential energy).
Hint: ex = 1 + x + + + + ···
2. A particle of mass m is acted on by a force whose potential energy is
constants positive , ; )( 32 babxaxxV −=
a) Graph this potential energy by hand (Computer use for checking ok)
b) Find the force
c) Discuss the effect of this force on the total mechanical energy of the particle.
d) Provide a qualitative discussion of the possible types of motion depending on the total
energy E. (bound, unbound motion, turning points, etc.)
e) Assume that the particle starts at the origin x = 0 with velocity v0. Show that if c
vv <
0
, where vc is a certain critical velocity, the particle will remain confined to a region near
the origin. Find vc.
x
2
2
x
3
2·3·
x
4
2·3·4
