Dr. Martin Hentschinski
martin.hentschinski@udlap.mx
curso: LFA3021 Mecánica Clásica (otoño 2023)
numero del ejercicio: 2
fecha: 22 de agosto de 2023
hora de asesorı́a: Friday, 10:00-11:00 CN124
Fecha lı́mite para la entrega de soluciones: August 30, 2023 in Blackboard.
Problem 1: We showed that a conservative force can be written as a gradient of a scalar
potential energy, Fi (~r) = − ∂r∂ i U (~r), i = x, y, z.
a) An important example of such a scalar potential energy is
GmM
U (~r) = − √
,
(1)
~r2
which is the potenial energy of a body of mass m moving in the gravitational field of another
body of mass M which is placed in the origin. Use Fi (~r) = − ∂r∂ i U (~r), i = x, y, z to determine
the corresponding conservative force.
Problem 2) A particle with mass m is moving periodically in a potential U (x) with turning
points x1 and x2 .
a) Using conservation of energy E, demonstrate that the period of the movement is given by
Z x2
√
dx
p
T = 2m
,
E − U (x)
x1
b) Explain why energy is conserved in this case.
Problem 3) This is an example with dissipative forces, i.e. resisistive forces. We consider
the case where the resisitive force in a certain medium (for instance air) is proportional to
the velocity of the particle, i.e. F~r = −k~r˙ . In the following we will consider a 1-dimensional
examples:
~ (t) = v(t)x̂ subject to the above resisa) find the displacement and the horizontal velocity v
~ (0) = 0 with velocity ~r˙ (0) = v0 x̂ and
tive force, if the particle is placed at time t = 0 at r
v0 > 0.
b) Determine the velocity of the particle in the limit t → ∞ as well as in the limit of vanishing
resistive force k → 0.
c) Determine the amount of the total energy of the particle which is lost due to friction for
t→∞
Comment: You get points if you present an argument, but for full points c) requires a calculation
Problem 4) We consider two particles with trajectories ~r1 (t) and ~r2 (t) and masses m1 and
m2 . To describe their motion, we introduce now
~r = ~r1 − ~r2
~ = m1~r1 + m2~r2
R
m1 + m2
1
(2)
a) Show that in absence of external forces, one has
~
~ 0) + V
~0 (t − t0 ),
R(t)
= R(t
(3)
~0 and position R(t
~ 0)
with an arbitrary intial velocity V
b) Show that
µ~r¨ = F~12 ,
and determine the parameter µ (which is called the reduced mass)
2
(4)