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Physics 325, Fall 2006
REVIEW
QUESTIONS
#1
and SOLUTIONS
1) A particle of mass
rn is moving
in the potential
i(r).:
ar2 + bra
Depending
on the sign of the constants
o and b, there are four cases
shown
in the
following
table
b>0 b<0
a)0
A
B
a<0
C
D
For
the four cases
in the table:
a) Sketch
the potentialU(r).
b) Find all the equilibrium
points of U (r). Specify whether
they are stable or un
stable.
c) Find the frequency
of small oscillations
around points of stable equilibrium.
I
o:
\ur
s\
\
o
d) In the above figure, the acceleration of the particle is plotted as afunction of
r f.or the four cases mentioned above. Match every acceleration graph to its
corresponding
potential.
e) Find the work done when the particle moves
from point p to point g as shown
in
the acceleration
graph (III).
lt
I
I
I
\/
OI
\
0
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View Full Document2) A cannon ball is frred uertically from ground
level at time t
:
0 with initial velocity u6.
The ball encounters
a force Fai. :
4v
due to air resistance
(v
is the velocity vector
for the ball and
4
is a constant).
Derive
the diffrential equation
for the velocity of the ball as a function of time for
all times before
the ball first hits the ground on return.
Solve
the equation derived
in (p). Determine
the terminal velocity of the ball u7.
Calculate
the height of the ball as.
a function of time.
Use the results of (b) ald (c) to show that the condition that the ball hits the
ground with u

u7 is that u6 D ur.
4) A rocket starts from rest at the center of a galaxy (r
0) at time t
0 and moves
radially outward following a straight line path. Gravitational and other forces cause
the rocket to move
in the spherically symmetric potential
u(r)
@
lro)2
where rn is the rocket mass (which is a function oi time) and 26, 76 &r€ some known
characteristic distance and time. In addition, the rocket engine
is fired at launch,
ejecting
fuel mass at the constant
rate a > 0 with speed u relative
to the rocket
frame.
All relativistic effects are
ignored
in this problem.
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 Fall '09
 Lamb
 mechanics, Mass

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