Vertical block-spring system differs to horizontal arrangement in the application of gravitational force. In horizontal orientation, gravitational force is perpendicular to motion and as such it is not considered for the analysis. In vertical orientation, however, the spring is in extended position due to the weight of the block before the block is set in SHM. It is in equilibrium in the extended position under the action of gravitational and spring force.
Clearly, the center of oscillation is the position of equilibrium. The block oscillates about the extended position – not about the position of neutral spring length as in the case of horizontal arrangement. Let us consider that the spring is extended by a vertical length “
y
0
y
0
” from neutral position when it is in equilibrium position. For a further extension “y” in spring, the spring force on the block is equal to the product of spring constant and total displacement from the neutral position,
F
=
-
k
y
0
+
y
F
=
-
k
y
0
+
y
Note that we have considered downward displacement as positive. The spring force acting upward is opposite to displacement and hence negative. In this case, however, the net restoring force on the block is equal to the resultant of spring force acting upwards and gravity acting downward. Considering downward direction as positive,
⇒
F
net
=
F
+
m
g
=
-
k
y
0
+
y
+
m
g
⇒
F
net
=
F
+
m
g
=
-
k
y
0
+
y
+
m
g
But, for equilibrium position, we have the following relation,
m
g
=
-
k
y
0
m
g
=
-
k
y
0
Substituting this relation in the expression of net restoring force, we have :
⇒
F
net
=
-
k
y
⇒
F
net
=
-
k
y
The important point to realize here is that net restoring force is independent of gravity. It is equal to differential spring force for the additional extension – not the spring force for the total extension from the neutral position. Now, according to Newton’s second law of motion, the net restoring force is equal to the product of mass of the block and acceleration,
⇒
F
net
=
m
a
=
-
k
y
⇒
F
net
=
m
a
=
-
k
y
⇒
a
=
-
k
m
y
⇒
a
=
-
k
m
y
This relation on comparison with SHM equation “
a
=
-
ω
2
x
a
=
-
ω
2
x
” yields same set of periodic expressions as in the case of horizontal block-spring arrangement :
⇒
ω
=
k
m
⇒
T
=
2
π
m
k
⇒
ν
=
1
2
π
k
m
⇒
ω
=
k
m
⇒
T
=
2
π
m
k
⇒
ν
=
1
2
π
k
m
In this case, however, we can obtain alternative expressions as well for the periodic attributes as spring force at equilibrium position is equal to the weight of the block,
m
g
=
-
k
y
0
m
g
=
-
k
y
0
Dropping negative sign and rearranging,
⇒
m
k
=
y
0
g
⇒
m
k
=
y
0
g
Hence, the alternative expressions of periodic attributes are :
⇒
ω
=
g
y
0
⇒
T
=
2
π
y
0
g
⇒
ν
=
1
2
π
g
y
0
⇒
ω
=
g
y
0
⇒
T
=
2
π
y
0
g
⇒
ν
=
1
2
π
g
y
0
Clearly, the extension of the spring owing to the weight of the block in vertical orientation has no impact on the periodic attributes of the SHM. One important difference, however, is that the center of oscillation does not correspond to the position of neutral spring configuration; rather it is shifted down by a vertical length given by :
⇒
y
0
=
m
g
k
⇒
y
0
=
m
g
k