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प्रश्न
Obtain the expression for the period of a simple pendulum performing S.H.M.
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उत्तर
Let m is Mass of the bob.
L is the Length of the massless string.
A free-body diagram of the forces acting on the bob.
θ is the angle made by the string with the vertical.
T is tension along the string.
g is the acceleration due to gravity.
∴ Restoring force, F = −mg sin θ ...(i)
As θ is very small (θ < 10°), we can write
sin θ ≅ θ° ∴ F ≅ mgθ
From the figure, the small angle `theta = x/L`
∴ `F = -mgx/L` ...(ii)
As m, g and L are constants, F ∝ −x
Thus, for small displacements, the restoring force is directly proportional to the displacement and is oppositely directed.
Hence, the bob of a simple pendulum performs linear S.H.M. for small amplitudes. The period T of oscillation of a pendulum from can be given as,
= `(2pi)/omega`
= `(2pi)/sqrt("acceleration per unit displacement")`
Using eq. (ii), `F = -mgx/L`
∴ `a = -gx/L`
∴ `a/x = -g/L = g/L` ...(in magnitude)
Substituting in the expression for T, we get,
`T = 2pi sqrt((L)/(g))` ...(iii)
The equation (iii) gives the expression for the time period of a simple pendulum.
संबंधित प्रश्न
Answer in brief.
State the law of simple pendulum.
A simple pendulum performs S.H.M. of period 4 seconds. How much time after crossing the mean position, will the displacement of the bob be one-third of its amplitude.
A simple pendulum of length 100 cm performs S.H.M. Find the restoring force acting on its bob of mass 50 g when the displacement from the mean position is 3 cm.
In a second’s pendulum, the mass of Bob is 50 g. If it is replaced by 100 g mass, then its period will be ______.
Obtain an expression for a time period of the simple pendulum.
The length of a simple pendulum is increased by 44%. The percentage change in its time period is ______.
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A person measures a time period of a simple pendulum inside a stationary lift and finds it to be T. If the lift starts accelerating upwards with an acceleration `("g"/3)`, the time period of the pendulum will be ______.
A pendulum performs S.H.M. with period `sqrt3` second in a stationary lift. If lift moves up with acceleration `"g"/3`, the period of pendulum is ______. (g = acceleration due to gravity)
Two simple pendulums of lengths 1.44 m and 1.96 m start swinging together. After how many vibrations will they again start swinging together?
The length of a seconds pendulum is 1m on the earth. If the mass and diameter of the planet is double than that of the earth, then the length of the second's pendulum on the planet will be ______
A simple pendulum has a length of 2m and a bob of mass of 100 grams. It is whirled in a horizontal plane. If the string breaks under a tension of 10 N, the angle made by the string with vertical is ______
(g = 10 m/s2)
The maximum kinetic energy of a simple pendulum is 'K'. Its displacement in terms of amplitude 'a', when kinetic energy becomes `(K/2)` is ______.
Define second’s pendulum.
The ratio of frequencies of two oscillating pendulums is 3 : 2. Their lengths are in the ratio.
Two weights w1 and w2 are suspended from the ends of light string over a smooth fixed pulley. If the pulley is pulled up with acceleration g, the tension in the string will be ______.
A clock regulated by a second pendulum keeps the correct time. During winter the length of the pendulum decreases by 1 %. How much will the clock gain or lose in one day?
The velocity of bob of a second’s pendulum when it is 6 cm from its mean position and amplitude of 10 cm, is ______.
A simple pendulum is used to investigate the dissipation of energy. Which quantity is plotted against time to analyse this dissipation?
