6.1 Basic Concepts and Formulae
237
Pendulums
Simple Pendulum (Small Amplitudes)
T = 2π
L
g
(6.16)
T is independent of the mass of the bob. It is also independent of the amplitude
for small amplitudes.
Seconds pendulum is a simple pendulum whose time period is 2 s.
Simple Pendulum (Large Amplitude)
For large amplitude θ 0 , the time period of a simple pendulum is given by
T = 2π
L
g
1 +
1
2
2
sin
2
θ 0
2
+
1.3
2.4
2
sin
4
θ 0
2
+
1.3.5
2.4.6
2
sin
6
θ 0
2
(6.17)
where we have dropped higher order terms.
Simple pendulum on an elevator/trolley moving with acceleration a. Time period
of the stationary pendulum is T and that of moving pendulum T .
(a) Elevator has upward acceleration a
T
= T
g
g + a
(6.18)
(b) Elevator has downward acceleration a
T
= T
g
g − a
(6.19)
(c) Elevator has constant velocity, i.e. a = 0
T
= T
(6.20)
(d) Elevator falls freely or is kept in a satellite, a = g
T
= ∞
(6.21)
The bob does not oscillate at all but assumes a fixed position.