1. The time period of a simple pendulum is
T = 2π √ (l/g)
⇒ T α √l or T α 1/√g
T α √ (l/g)
Using these relations. We may conclude
⇒ The graph between
T2 and l is a straight line.
⇒ The graph between
T and l is a parabola.
⇒ The graphs l - T
and l - T2 intersect at T = 1s
⇒ The graph between
T2 and 1/g is a straight line.
⇒ The graph between
T2 and g is a rectangular hyperbola.
⇒ Finally, the
graph between T and √(l/g) is also a straight line.
2. In the case of water oscillating in a U - tube
T = 2p√(l/g)
Where, h is the height of liquid column in
each limb.
When a ball of mass m is made to oscillate in
the nech of an air chamber having volume V and neck area A, then
3. When a pendulum is kept in a car which is
sliding down, then
Where, θ is the angle of inclination
4. If a simple pendulum oscillates in a
non-viscous liquid of density ρ, then its time period is
ρ = density of suspended mass.
5. If the mass m attached to a spring oscillates
in a non-viscous liquid of density σ, then its time period is
Where, k = force constant.
6. If a small ball is rolling down in
hemispherical bowl. Time period,
Where, R = radius of the bowl and
r = radius of the ball
7. For a body executing SHM in a tunnel dug
along any chord of earth.
Time period, =84.6 min
Where Re is the radius of earth.
8. If the time period of simple pendulum is 2s,
then it is called as second’s pendulum
9. If the simple pendulum is placed in some
non-inertial frame of reference like an accelerated lift, g is replaced by geff
whose value can be computed by considering the inertial force. In these
cases the equilibrium position may also change.
10. If the length of simple pendulum is very
large, then g can’t be taken along vertical direction.
In this case,
Where, R = Radius of length of the pendulum.
11. If temperature of system changes, then time
period of simple pendulum changes due to change in length of the simple
pendulum.
12. If a simple pendulum is in a carriage which
is accelerating with an acceleration a, then
geff = g - a
eg, if the acceleration a is upwards, then
If the acceleration a is downwards, then (g > a)
|geff| = g + a and
If the acceleration a is in the horizontal
direction, then
|geff| = √ (a² + g²)
In a freely falling lift, geff = 0 and T = ∞ i.e., the pendulum will not
oscillate.
13. If in addition to gravity one additional
force F (e.g., electrostatic force Fe) is also acting on the both,
then in that case
geff = g + F/m
Here, m is the mass of the bob.
Torsional
Pendulum: in a torsional
pendulum, an object is suspended from a wire. If such a wire is twisted due to elasticity,
it exerts a restoring torque τ = Cθ
In this case, time period is given by
Where, l = moment of inertia of the object
C = torsional constant of wire =
η = modulus of elasticity of wire
r = radius of wire
l = length of wire
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