Advertisements
Advertisements
प्रश्न
Consider a pair of identical pendulums, which oscillate with equal amplitude independently such that when one pendulum is at its extreme position making an angle of 2° to the right with the vertical, the other pendulum makes an angle of 1° to the left of the vertical. What is the phase difference between the pendulums?
Advertisements
उत्तर
Consider the situations shown in the diagram (i) and (ii)
Assuming the two pendulums follow the following functions of their angular displacements
`θ_1 = θ_0 sin(ωt + phi_1)` .......(i)
And `θ_2 = θ_0 sin(ωt + phi_2)` .......(ii)
As it is given amplitude and time period are equal but phases being different.
Now, for the first pendulum at any time t
`θ_1 = + θ_0` .....[Right extreme]
From equation (i), we get
⇒ `θ_0 = θ_0 sin(ωt + phi_1)` or 1 = `sin(θt + phi_1)`
⇒ `sin pi/2 = sin(ωt + phi_1)`
or `(ωt + phi_1) = pi/2` ......(iii)
Similarly, at the same instant t for pendulum second, we have `θ_2 = - θ_0/2` where θ0 = 2° is the angular amplitude of the first pendulum. For the second pendulum, the angular displacement is one degree, therefore `θ_2 = θ_0/2` and a negative sign is taken to show for being left to mean position.
From equation (ii), then `- θ_0/2 = θ_0 sin(ωt + phi_2)`
⇒ `sin(ωt + phi_2) = - 1/2`
⇒ `(ωt + phi_2) = - pi/6 or (7pi)/6`
or `(ωt + phi_2) = - pi/6 or (7pi)/6` ......(iv)
From equations (iv) and (iii), the difference in phases
`(ωt + phi_2) - (ωt + phi_2) = (7pi)/6 - pi/2`
= `(7pi - 3pi)/6`
= `(4pi)/6`
Or `(phi_2 - phi_1) = (4pi)/6 = (2pi)/3` = 120°
APPEARS IN
संबंधित प्रश्न
When the length of a simple pendulum is decreased by 20 cm, the period changes by 10%. Find the original length of the pendulum.
A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.
Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
let us take the position of mass when the spring is unstretched as x = 0, and the direction from left to right as the positive direction of the x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is
(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?
The acceleration due to gravity on the surface of moon is 1.7 ms–2. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s? (g on the surface of earth is 9.8 ms–2)
Answer the following questions:
A time period of a particle in SHM depends on the force constant k and mass m of the particle: `T = 2pi sqrt(m/k)` A simple pendulum executes SHM approximately. Why then is the time
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?
Define practical simple pendulum
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?
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 ______.
A particle executing S.H.M. has a maximum speed of 30 cm/s and a maximum acceleration of 60 cm/s2. The period of oscillation is ______.
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?
The length of a second’s pendulum on the surface of earth is 1 m. What will be the length of a second’s pendulum on the moon?
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 simple pendulum of time period 1s and length l is hung from a fixed support at O, such that the bob is at a distance H vertically above A on the ground (Figure). The amplitude is θ0. The string snaps at θ = θ0/2. Find the time taken by the bob to hit the ground. Also find distance from A where bob hits the ground. Assume θo to be small so that sin θo = θo and cos θo = 1.
A particle at the end of a spring executes simple harmonic motion with a period t1, while the corresponding period for another spring is t2. If the period of oscillation with the two springs in series is T, then ______.
