Given that the length of pendulum increases by 1%. Now we need to find out how many seconds will a pendulum lose or gain per day Equation of time period of pendulum is that : T = 2π√
L
g
= 2Ï€(
L
g
)
1
2
where : T is the time period , L is length of pendulum and g is constant Differentiate this equation with respect to L :
d
dL
(T) =
d
dL
[2π√
L
g
] =
d
dL
[2Ï€(
L
g
)
1
2
] = 2π×
d
dL
(
L
g
)
1
2
= 2π×
1
2
×(
L
g
)−
1
2
×
1
g
=
Ï€
g
×
1
√
L
g
(
dT
dL
)2 =
Ï€2
g2
×
1
L
g
=
Ï€2
gL
dT
dL
=
Ï€
√gL
dT =
Ï€
√gL
×dL Given that the length of pendulum increases by 1%
dL
L
= 1% =
1
100
= 0.01 dL = 0.01 L ∴ dT =
Ï€
√gL
× 0.01 L dT =
0.01Ï€
√g
×
L
√L
=
0.01Ï€
√g
×
√L×√L
√L
=
0.01π√L
√g
= 0.01π√
L
g
= 0.01 ×
1
2
×2π√
L
g
= 0.005 × T Total time per day T = 24 × 60 × 60 = 86400 dT = 0.005 × 86400 = 432 s ∴ Time lost per day , dT = 432 s