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Examination of Scientific culture
Physics
Exercise 1
Exercise of mechanics (recommended for students having Physics
as a secondary subject)
We consider the following situation of an open hourglass
suspended by a spring :
At time t = 0, the hourglass is released without velocity from a
position x0 away from itsequilibrium position xe and the aperture
of the hourglass is opened.
Q1. One assumes that the motion of the hourglass does not affect
significantly the escape ofsand, which behaves as if the hourglass
were at rest. We define m(t) as the mass of the hour-
glass at time t. What can you assume about the mass flow rate D
=dm
dt? Can you propose an
experiment to test this assumption ?
Q2. Let us define m0 the initial mass of the hourglass. Moreover
one assumes its mass whenit is empty is negligible. One defines a
vertical axis Oz, O being the position of the end of thespring when
it is unloaded. k is the spring constant. What is the equilibrium
position of the
hourglass xe(t) when there are no oscillations ?
Q3. Let q be the velocity of the sand with respect to the bottle
aperture and x(t) the distance
between the hourglass and its equilibrium position. What is the
momentum changedp
dtof the
hourglass between times t and t + dt ?
Q4. Justify the assumption :dp
dt m(t)
d2x
dt2,
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Q5. Using the fundamental principle of mechanics and making the
assumptions that you findnecessary, show that x(t) verifies the
following differential equation :
m(t)d2x
dt2+
dx
dt+ kx = 0, (3)
where is the friction coefficient of air on the hourglass.It is
not easy to directly integrate the previous equation because of the
time varying term m(t).In order to have a physical insight of the
motion one neglects the equilibrium position changexe(t) and one
assumes that at each moment the motion of the hourglass is
sinusoidal :
x(t) = A(t)sin[(t)t + (t)],
with (t) =
k
m(t). A(t) and (t) are slowly varying quantities.
Q6. Express the potential energy of the system Ep(t) at time t
as a function of x(t). Showthat one has :
dEpdt
=
dx
dt
2m(t)
dx
dt
d2x
dt2
Q7. Evaluate the kinetic energy change dEcdt
as a function ofm(t), D and the derivatives ofx(t).
Q8. Show that the mechanical energy of the hourglass Em(t)
verifies :
dEmdt
=
+
D
2
dx
dt
2(4)
Q9. One considers one period of the pendulum around time t. What
is then the total mecha-
nical energy as a function of A(t) ?
Q10. Evaluate the energy loss term, right hand side of equation
(4), over one oscillation as afunction of A(t).
Q11. Infer from the previous questions a differential equation
on A(t).
Q12. Show that the solution to this equation can be written as
:
A(t) = A(0)
1
Dt
m0
2D
+1
4
.
Q13. Discuss the appearance of function A(t) as a function of
the relative values of parame-ters and D.
Q14. Demonstrate that, in the regime where the friction is
dominant over the mass loss, oneretrieves the usual results of the
damped oscillator.
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Examination of Scientific culture
Physics
Exercise 2
Electrostatics Exercise
This exercise consists of three parts. Topic-wise they are
related, but it is nevertheless possibleto treat the first two
independently. The third part consists of a summary of the
physicalphenomenon studied in the first two parts.
1.) A model of the hydrogen atom
In this part we want to find the charge distribution creating
the potential
V(r) =q
40
1
re
r
a (5)
1.a) Calculate the associated electric field. Express the flux
of the electric field through a sphereof radius r.1.b) Take the
limits r going to 0 and to infinity of the expression of the flux.
What can oneconclude from them?1.c) Calculate the charge density
(r) associated with the flux.1.d) The potential constitutes a
quantum model for the electrical field of a proton in the pre-sence
of an electron (hydrogen atom). What is the probability density of
the electron (defined
by p(r) = dqdr where dq is the charge within the spherical shell
(r,r+dr)) ? What is the role playedby a for the probability density
of the electron ? Give an order of magnitude for a.1.e) Justify the
expression screened Coulomb potential associated with the potential
above.1.f) Propose a modification of the theory for describing the
Helium atom.1.g) Does an analogous model exist for the
gravitational case ?
2.) Screening
In this part of the exercise we study a model of a plasma. We
consider N negatively chargedparticles (charge q) in a box of
volume V. We take V and N big such as not to worry about
boundaries nor about the granularity of the particles. We
suppose moreover the presence of auniform positive charge Nq/V
everywhere in the box.2.a) What is the role of the uniform positive
charge ? Give the total charge density (r) at everypoint in
space.2.b) We introduce now a point charge Q > 0 at the origin,
and we want to calculate the varia-tions of the charge density and
of the potential induced by Q.(i) Using Boltzmanns probability
distribution and a mean field approximation, give the densityas a
function of the electrical potential, when the plasma is in
equilibrium at temperature T.(The mean field approximation consists
in assuming that each particle sees the potential wi-thout
perturbing it.)(ii) Find a differential equation for the
potential.
(iii) Under which condition can one linearize the equation ?(iv)
Assuming this condition satisfied, solve the linearized equation.
Define a characteristic lengthand interpret it physically.
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3.) Summary
Summarize in a few sentences the essential points of the
phenomenon of screening, based on theresults of parts 1.) and
2.).
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1.e) Compute the thermal transfer QC received by the air through
the room and the workWCp received by the air inside the compressor.
Deduce the efficiency e of the heat pump.
1.f) What is the efficiency of an ideal Carnot heat pump ec,
working between the same heatsources ? Comment.
2.) Let it be a semi-infinite gaseous column (z > 0), the
boundaries of the column imposea given constant temperature T. The
column is under the influence of the uniform gravity fieldgz. The
density of the gas at z = 0 is 0. If one introduces an object of
mass M and volumeV :
2.a) find the equilibrium height of the object.2.b) Find the
frequency of oscillation of the object around its equilibrium
position.2.c) Now we change the boundaries of the column by an
adiabatic insulator, what is the newequilibrium height of the
object ?2.d) Quick question : What is the height of the atmosphere
?
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