Laws of pendulum
From the expression for the time period of oscilations of a pendulum the following laws are enunciated.
( i) The law of length
The period of a simple pendulum varies directly as the square root of the length of the pendulum. (i.e) T α l
(ii)The law of acceleration
The period of a simple pendulum varies inversely as the square root of the acceleration due to gravity.
(i.e) T α 1 g
(iii) The law of mass
The time period of a simple pendulum is independent of the mass and material of the bob.
(iv)The law of amplitude
The period of a simple pendulum is independent of the amplitude provided the amplitude is small.
Note : The length of a seconds pendulum is 0.99 m whose period is two seconds.
2 = 2π.rt(l/g)
l= 9.81x 4 / 4 π2=0.99 m
Oscillations of simple pendulum can also be regarded as a case of angular SHM.
Let θ be the angular displacement of the bob B at an instant of time. The bob makes rotation about the horizontal line which is perpendicular to the plane of motion as shown in Fig..
Restoring torque about O is τ = − mg l sin θ
τ = −m g l θ [ ∵ θ is sm all] ...(1)
Moment of inertia about the axis = m l 2 ...(2)
If the amplitude is small, motion of the bob is angular simple harmonic. Therefore angular acceleration of the system about the axis of rotation is
α = r/l = -mgl θ / ml2
α = - (g/l) θ ?.(3)
We know that α = −ω 2 θ ??.(4)
Comparing (3) and (4)
−ω 2θ = -(g/l) θ
angular frequency ω = rt(g/l)
Time period T = = 2π / ω = 2π .rt(l/g)
Frequency n = 1 / 2 π . rt(g/l)