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Technical University of Sofia
Branch Plovdiv
Theoretical Mechanics
DYNAMICS
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Lecture 2
Special Cases. The force is determined by the time.
mx  F  t 
t
F t 
dx
1
m  F  t   dx 
dt  x   F  t  dt
dt
m
m0
t
t t
1 t


dx 1
1 
  F  t  dt  dx    F  t  dt  dt  x     F  t  dt  dt
dt m 0
m 0 0
m 0


Lecture 2
Special Cases. The force is determined by the time.
Sample Problem 4
The bullet of mass m is given a velocity due to gas pressure caused by
the burning of powder within the chamber of the gun. Assuming this
pressure creates a force of F = F0sin(πt / t0) on the bullet, determine
the velocity of the bullet at any instant it is in the barrel. What is the
bullet’s maximum velocity? Also, determine the position of the bullet in
the barrel as a function of time.
Lecture 2
Special Cases. The force is determined by the velocity.
mx  F  v 
v
dv F  v 
dv
dt
dv



 t  m
  v
dt
m
F v m
F v 
v0
t    v   v   1  t 
t
dx
v
  1  t   dx   1  t  dt  x    1  t  dt
dt
0
Lecture 2
Special Cases. The force is determined by the velocity.
Sample Problem 5
A car of mass m is traveling at a slow velocity v0. If it is subjected to
the drag resistance of the wind, which is proportional to its velocity, i.e.,
FD = kv. Determine the distance and the time the car will travel before
its velocity becomes 0,5 v0. Assume no other frictional forces act on
the car.
Lecture 2
Special Cases. The force is determined by the position.
mx  F  x 
x
F  x
dv dv dx
dv F  x 
v2 1

v

 vdv 
dx 
  F  x  dx 
dt dx dt
dx
m
m
2 m x0
x
2
v    x 
F  x  dx

m x0
x
dx
dx
dx
v
   x 
 dt  t  
dt
  x
  x
x0
Lecture 2
Special Cases. The force is determined by the position.
Sample Problem 6
The spring of constant k=200N/m is attached to both the support and
the 2-kg cylinder, which slides freely on the horizontal guide, If a
constant 10-N force is applied to the cylinder at time t=0 when the
spring is undeformed and the system is at rest, determine the velocity
of the cylinder when x=40 mm. Also determine the maximum
displacement of the cylinder.