Advertisements
Advertisements
Question
Obtain the expression for the period of a simple pendulum performing S.H.M.
Advertisements
Solution
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.
RELATED QUESTIONS
Define an ideal simple pendulum.
Answer in brief.
State the law of simple pendulum.
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.
What is a second pendulum?
Obtain an expression for a time period of the simple 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)
Work done in stretching a wire through 1 mm is 2 J. What amount of work will be done for elongating another wire of same material, with half the length and double the radius of cross-section, by 1 mm?
A simple pendulum has a time period 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 T2 / T1 is ______.
Two pendulums begin to swing simultaneously. The first pendulum makes nine full oscillations when the other makes seven. The ratio of the lengths of the two pendulums is ______.
Two light balls are suspended as shown in figure. When a stream of air passes through the space between them, the distance between the balls will ______.
If a pendulum which gives correct time beats second on ground at a certain place is moved to the top of a tower 320 m high, then the loss of time of the pendulum clock in one day, in second, is ______.
Two simple pendulums A and B are made to oscillate simultaneously and it is found that A completes 10 oscillations in 20 sec and B completes 8 oscillations in 10 sec. The ratio of the lengths of A and B is ____________.
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)
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 ______.
Derive a formula for the length of second's pendulum.
The bob of a simple pendulum performs SHM with period T in air and with period T1 in water Relation between T and T1 is ______.
(neglect friction due to water, density of the material of the bob is = `9/8xx10^3` kgm3, density of water = 1 gcc-1)
A pendulum bob is swinging in a vertical plane, such that its angular amplitude is less than 90°. At its highest point, the string is cut. Which trajectory is possible for bob afterwards?
A uniform metal rod is used as a bar pendulum. If the room temperature rises by 10°C and coefficient of linear expansion of the metal of the rod is 2 × 106 °C-1, the period of pendulum will increased by ______.
A simple pendulum of length 196 cm performs linear SHM of amplitude 8 cm. Find the restoring force on its bob of mass 50 g at an extreme position.
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?
A simple pendulum is used to investigate the dissipation of energy. Which quantity is plotted against time to analyse this dissipation?
A student measures time for 20 oscillations of a simple pendulum as 30 s, 32 s, 35 s and 35 s. If the minimum division in the measuring clock is 1 s, then correct mean time (in second) is ______.
