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A mass is hooked on a flexible spring suspended vertically from
a rigidsupport. Suppose the amount of elongation on the spring
until it attain
equilibrium is s. Then at this point, the weight of the mass
will have the same
magnitude as the restoring force. Once we pull the spring from
the point of
equilibrium and release it, then a restoring force will be
applied on it.rovided that there is no other external force acting
on the system ! i.e.
retarding force", then we can equate the force acting on the
spring and therestoring force. #esides, without damping ! retarding
force" the spring$ mass
system will continue to oscillate continuously.
Since , so
So we will obtain the following expression for the force acting
on the spring%
.
The usual case for the spring mass system is%
&" 'e pull the mass downwards ! the spring and release.
(" 'e push the mass upwards !the spring compresses"
mg ks= -mx ks= -&&
kx mg- +
mx kx= -&&
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Solving the second order O)* !",
#y using
& & ( (x c y c y= +
mx kx= -&&
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+irst try%
#y substituting the expression into the equation , we
obtain%
Thus the two roots of a is%
(a m k= -
aty e=
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, but let
So our solution is for the (nd order O)* is%
onsider case &"
- -
& (
i t i
x c e c e-
= +
- k
m=
ka im
= \
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f the initial condition is such that , then , so
.
#esides, if we start the whole spring mass oscillation system
such that .
!/" /x =( ) ( & &cos- sin - cos - sin - c t i t c t i =
+ - -
& (c c= -
!/" /x =max0!/"x v=
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Since , .
Thus.
max&
(-
vc
i=
&0! " (-cos-x t c i =
&( sin-x c i t=
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So the solution is% .
onsider case ("
f the initial condition is such that , then
max! " sin--
vx t =
& (c A c= -!/"x A=
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t is clear that when its position is at its amplitude then the
velocity of the
block is /.
Since , we get%
So
0!/" /x =
& (c c=
- -
& (0- i t i tx ic e ic e-= -
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An alternative method can be used%
#ut in real life situations, we usually do not have free motion
! i.e. retarding force isalways present". 1et2s only consider the
first order resistive force which is .
So the force
( ) ( cos- sin - cos- sin -( (cos-
A A t i t t i
A t
= + + -
=
retardingF Cx= -
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( ) ( )
( ) ( )
( )
( ) ( )
- -
& (
& (
& ( & (
& (
( ( & & (
(
&
(
((( (
& ( & ( & ( & (
cos- sin - cos- sin -
cos- sin -
cos- sin -
sin- 3
,
)sin3
cos3
n3
4 (
i t i t
c e c ec t i t c t i t
c c t i t c c
A t A t
D tA
c A c ci
D
A
A
A A c c i c c c c i
-= +
+ - -
- + +
+
+
- = + =
=
=
=
= + = - + + = - =
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& (
&
&
(
(
& (
( 5( (
& (
5( (
( 5( (
& (
/
5 - /
5 - /
5 5 4-
(! "
4- 5 4- /, , ! "
(
4- /, ! "
5 4- 4- /, , ! "
(
a t a t
a t
t
a t a
x Cx kx
x x
a
t c e c e
ia x t e c e c e
x t ce
a x t e c e c e
-
-
-
+ + =
+ + =
+ + =
- -=
= +
-- < = = +
- = =
- - > = = +
&& &
&
