Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.
DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.
We introduce 100ml of water in a U-shaped tube of section Su2004=u20041 cm²
(Figure). On a side of the tube, we introduce 100ml of oil with a
density ρ_(x) equal to 80% of the one of water (ρ_(w)). Find the
difference of level between the two sides of the tube.For a homogeneous fluid in equilibrium the level surfaces are also
isobars. We thus have p_(B)u2004=u2004p_(C), as well as p_(A)u2004=u2004p_(D)u2004=u2004p_(atm).
Moreover:p_B &=& p_A + \rho_x \,g\, h~,\\
p_C &=& p_D + \rho_w \,
The stationary flow of a fluid is described by the field of velocities
v⃗u2004=u2004ωru2006e⃗_(ϕ). Determine the acceleration of a fluid.
In a stationary flow, the acceleration of a fluid particle is :
a=(v⋅∇)v. For this problem we will use
cylindrical coordinates. So:=\left(\o
Prove the expressions for the height of the center of mass of the cone,
respectively the half-sphere, given in the previous problem’s hint.
We will rigorously prove the calculation for the center of mass of the
cone, respectively, the half-sphere, in the previous problem. The center
of mass of a solid is given by
rG=M1∫volumerρ(r)d3r.
In our case, the solids are homogeneous, therefore ρ(r⃗)u2004=u2004ρ. For both
the cone and the half-sphere, the volume int
In the plane yz a infinite conducting plate carries a surface current
density that is uniform K=Kz^. Calculate the
magnetic field produced by this current in a point P(x,0,0) of the x
axis, by using : a) The Ampere law; b) the Biot-Savart law.a) By symmetry, we have: B(x,0,0)=By(x)y^.
Indeed, the Biot-Savart law shows that B is perpendicular to K.
Moreover, B_(x)u2004=u20040, because the inversion of the current direction (or
if we turned with 180 degr
xa0Determine the position in the plane of the center of mass of this system
consisting of the three square plates, as represented of the figure
hereby. The square plates are homogeneous and their edges have a length
a.The two light plates have the same mass m, the mass of the dark plate
is the double.xa0We are in the coordinate system (O,e_(x),e_(y)) of the sketch hereby. By
symmetry, the centre of mass of each square plate is at its centre : so
at the points
$\left(\frac{a}{2}, \frac{a}{2}\right) , \
Let us consider a flat disk of radius R carrying an uniformally
distributed charge Q. The surface charge density is σ.Calculate the electric field and the potential in a point P on the axis
of the disk as function of the distance z between P and the disk (Use
the results of exercise 1).Deduce the electric field created by the plate of infinite dimension
with the same charge density σ (make R tend to infinity).In the case where R is finite, discuss the limit cases zu2004≫u2004R and zu2004≪u2004R.
We us
When Firmin, the house painter, is in the middle of his nacelle and the
left dynamometer indicates 600xa0N, what does the right dynamometer show?If he’s not in the middle of the nacelle and the left dynamometer
indicates 400xa0N, what does the right dynamometer then show?Since the cable is stuck, Firmin decides to get out of his nacelle, from
the right. Just before Firmin lets go of the right cable, the left
dynamometer indicates 200xa0N. What does the right dynamometer show now?And now, find Firmin’s
A ladder with mass m and length L is resting on a vertical wall. The
angle between the wall and the ladder is α and the friction coefficient
between the ladder’s foot and the floor isxa0μ. There is no friction
between the wall and the ladder.What’s the maximal value αmax angle α can take such that
the ladder’s bottom doesn’t slip on the floor and fall?
We consider the ladder as the system, seen from the referential of the
floor and the wall. We choose a Cartesian reference frame with a
horiz
A wire of length L and of constant line charge density λ is vertically
aligned between z=−2L and z=2L. Calculate the
electric field E at point Pu2004=u2004(r,0,0). Check the limits Lu2004≫u2004r and
Lu2004≪u2004r.
By the symmetry of the problem, the created field in P(r,0,0) is radial:
E=Err^. The contribution of an infinitesimal element dz
of charge dq at point P(r,0,0) is:
$$dE_{r}=\frac{1}{4 \pi \epsilon_{0}}\frac{dq}{{\mbox{$\resizebox{.16in}{.08in}{\includeg
We consider a square plate with side of length a and uniform density. We
pierce a circular hole of radius R in the plate such that the distances
between the hole’s edge and two adjacent edges of the plate are b and c,
with b,u2006cu2004
A yoyo is made up of a disk of radius R and mass M, around which a
massless thread is wound up. The thread is attached to the ceiling and
remains constantly vertical; gravity acts on the yo-yo which is free to
lower down.
Compute the disk’s moment of inertia I around an axis perpendicular
to the disk’s plane and going through its center.
Find the acceleration a⃗ of the yo-yo’s center of mass, as well as
the thread’s tension T⃗.
A disk’s moment of inertia with respect to its revolution
The inclined pendulum consists of a material point of mass m that moves
without friction along a circle of radius R on an inclined plane. α is
the angle between the inclined plane and the horizontal plane.
Represent graphically the pendulum, the chosen coordinate system and
the forces.
Determine the equations of motion of the material point.
System and frame: The system is the pendulum mass m. We take a
cylindrical frame (O,e_(ρ),e_(ϕ),e_(z)).Constraints: The mass m has to move along a circ
A closed box of mass M is resting on a table. Inside the box is a marble
of mass m, attached to the lid by a spring with spring constant k and
rest length l₀. What’s the minimal amplitude A of the marble’s vertical
oscillation for the box to lift from the table? We assume l₀ and k to be
large enough for the marble to avoid hitting the lid or the table before
the box takes off.u2005Indication: treat both the box and the marble as material points.
The box will take of if its bonding force with the table
A wire carrying a uniform line charge density λu2004>u20040 has been folded as
shown in the figure. Calculate the electric field at the point O and the
potential with V(∞)u2004=u20040.The potential at the point O is obtained by decomposing the
contributions of the two segments and of the half circle. With V(∞=0) we
have (be careful at the order of limits of the integrales): \frac{1}{{4\pi \varepsilon _0 }}\left(~ {\int\limits_
A car’s suspension is comfortable when it has a vertical oscillation
period that our bodies are used to: the oscillation period of walking,
Tu2004=u20040.8u2005s.
Find the spring constant k of the suspension in a car of mass
M=1540kg.
The car lowers when we place a trunk of mass m=70kg in
it. Find its vertical displacement.
To make the car more comfortable, we add a shock absorber/dampener that
creates a friction force F⃗u2004=u2004u2005−u2005bv⃗_(z), where v⃗_(z)
The current through the coil on the right in Fig.xa0fig:fig7
varies as shown in Fig.xa0fig:fig7a. Which graph in
Fig.xa0fig:fig7b best shows the ammeter reading as a function of
time?The field created by the first loop:
first, it constantly grows with time ⟹ should induce a constant
current in the second loop;
then it’s constant ⟹ no current in the second loop;
finally, it constantly drops with time ⟹ should induce a constant
current in the opposite directi
A student of the general physics course takes part in a snowball fight
with a friend. This friend manages to catch the balls and to send them
back immediately.The student knows that for a snowball to reach a specific impact point
with a given initial speed, it can follow two different trajectories
corresponding to different throwing angles and different time of
flights. Thus, in order to win the game, the student decides to throw
two snowballs, at different times and with different throwing angles.
The ba
Consider the volume density of charge ρ(x,y,z)u2004=u2004ρ(x) of the figure on
the left, infinite in the directions y and z, where: aρ₀₊u2004=u2004Au2006[Cu2006m⁻²].
a) Determine and graphically represent the electric field associated for
all of the values of x. Determine and graphically represent the
potential V(x), with the condition V(−a/2)u2004=u20040.b) Now consider the charge density ρ(x,y,z)u2004=u2004ρ(x) of the figure on the
right side, where bρ⁰⁻u2004=u2004aρ₀₊u2004=u2004A. Determine and graphi
