264
6 Oscillations
(b) The time period is given by
T =
2π
ω
= 2π
h
2g
6.26 (a) Let the gas at pressure P and volume V be compressed by a small length
x, the new pressure being p and new volume V (Fig. 6.18) under isothermal conditions.
P
V
= PV
or P
(l − x)A = Pl A
where A is the cross-sectional area.
P
=
Pl
l − x
− P
1 −
x
l
−1 P
1 +
x
l
where we have expanded binomially up to two terms since x << l. The
change in pressure is
P = P
− P =
Px
h
The unbalanced force
F = −P A = −
APx
l
and the acceleration
a =
F
m
= −
APx
ml
= −ω
2 x
which is the equation for SHM.
Fig. 6.18
6 Oscillations
(b) The time period is given by
T =
2π
ω
= 2π
h
2g
6.26 (a) Let the gas at pressure P and volume V be compressed by a small length
x, the new pressure being p and new volume V (Fig. 6.18) under isothermal conditions.
P
V
= PV
or P
(l − x)A = Pl A
where A is the cross-sectional area.
P
=
Pl
l − x
− P
1 −
x
l
−1 P
1 +
x
l
where we have expanded binomially up to two terms since x << l. The
change in pressure is
P = P
− P =
Px
h
The unbalanced force
F = −P A = −
APx
l
and the acceleration
a =
F
m
= −
APx
ml
= −ω
2 x
which is the equation for SHM.
Fig. 6.18