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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 ______.
What is a second pendulum?
A simple pendulum of length L has mass m and it oscillates freely with amplitude A. At the extreme position, its potential energy is (g = acceleration due to gravity)
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Define second’s pendulum.
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(N = maximum electron density of lonosphere, g = acceleration due to gravity)
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(neglect friction due to water, density of the material of the bob is = `9/8xx10^3` kgm3, density of water = 1 gcc-1)
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 ______.
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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 ______.
At the highest point of its swing, a simple pendulum has maximum potential energy and minimum kinetic energy. What is the total mechanical energy of the pendulum at any point during its motion?
If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is `x/2` times its original time period. Then the value of x is ______.
