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प्रश्न
Answer the following questions:
The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that T is greater than `2pisqrt(1/g)` Think of a qualitative argument to appreciate this result.
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उत्तर १
In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as:
F = –mg sinθ
Where,
F = Restoring force
m = Mass of the bob
g = Acceleration due to gravity
θ = Angle of displacement
For small θ, sinθ = θ
For large θ, sinθ is greater than θ.
This decreases the effective value of g.
Hence, the time period increases as:
`T = 2pi sqrt(1/g)`
Where, l is the length of the simple pendulum
उत्तर २
The restoring force for the bob of the pendulum is given by
`F = -mg sintheta`
if `theta` is small thensin `theta = theta = y/l` `:. F = -(mg)/l y`
i.e the motion is simple harmonic and time period is` T = 2pi sqrt(1/g)`
Clearly, the above formula is obtained only if we apply the approximation `sin theta ~~ theta`
For large angles this approximation is not valid and T is greater than `2pi sqrt(1/g)`
संबंधित प्रश्न
The period of a conical pendulum in terms of its length (l), semi-vertical angle (θ) and acceleration due to gravity (g) is ______.
When the length of a simple pendulum is decreased by 20 cm, the period changes by 10%. Find the original length of the pendulum.
The phase difference between displacement and acceleration of a particle performing S.H.M. is _______.
(A) `pi/2rad`
(B) π rad
(C) 2π rad
(D)`(3pi)/2rad`
Answer the following questions:
A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?
A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?
A clock regulated by seconds pendulum, keeps correct time. During summer, length of pendulum increases to 1.005 m. How much will the clock gain or loose in one day?
(g = 9.8 m/s2 and π = 3.142)
Show that motion of bob of the pendulum with small amplitude is linear S.H.M. Hence obtain an expression for its period. What are the factors on which its period depends?
Show that, under certain conditions, simple pendulum performs the linear simple harmonic motion.
If the particle starts its motion from mean position, the phase difference between displacement and acceleration is ______.
A simple pendulum has a time period of T1 when on the earth's surface and T2 when taken to a height R above the earth's surface, where R is the radius of the earth. The value of `"T"_2 // "T"_1` is ______.
If the maximum velocity and acceleration of a particle executing SHM are equal in magnitude, the time period will be ______.
Two identical springs of spring constant K are attached to a block of mass m and to fixed supports as shown in figure. When the mass is displaced from equilibrium position by a distance x towards right, find the restoring force
When will the motion of a simple pendulum be simple harmonic?
Find the time period of mass M when displaced from its equilibrium position and then released for the system shown in figure.
A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period. `T = 2πsqrt(m/(Apg))` where m is mass of the body and ρ is density of the liquid.
A tunnel is dug through the centre of the Earth. Show that a body of mass ‘m’ when dropped from rest from one end of the tunnel will execute simple harmonic motion.
A pendulum of mass m and length ℓ is suspended from the ceiling of a trolley which has a constant acceleration a in the horizontal direction as shown in the figure. Work done by the tension is ______.
(In the frame of the trolley)
