Advertisements
Advertisements
प्रश्न
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?
Advertisements
उत्तर १
(a) x = 2sin 20t
(b) x = 2cos 20t
(c) x = –2cos 20t
The functions have the same frequency and amplitude, but different initial phases.
Distance travelled by the mass sideways, A = 2.0 cm
Force constant of the spring, k = 1200 N m–1
Mass, m = 3 kg
Angular frequency of oscillation:
`omega = sqrt(k/m)`
`= sqrt(1200/3)= sqrt400 = 20 rad s^(-1)`
a) When the mass is at the mean position, initial phase is 0.
Displacement, x = Asin ωt
= 2sin 20t
b) At the maximum stretched position, the mass is toward the extreme right. Hence, the initial phase is `pi/2`
Displacement , `x = Asin(omegat + pi/2)`
`= 2sin (20t + pi/2)`
= 2cos 20t
(c) At the maximum compressed position, the mass is toward the extreme left. Hence, the initial phase is `(3pi)/2`
Displacement, `x = Asin(omegat + 3pi/2)`
`= 2sin (20t + 3pi/2) = - 2 cos 20 t`
The functions have the same frequency (`20/(2pi) Hz`) and amplitude (2 cm), but different initial phases `(0, pi/2, (3pi)/2)`
उत्तर २
a =2 cm, omega = `sqrt(k/m) = sqrt(1200/3) s^(-1)= 20s^(-1)`
a) Since time s measured from mean position
b) At the maximum stretched position, tyhe body is at the extreme right position. The initial phase is `pi/2`
`:. x = a sin (omegat + pi/2) = a cos omegat = 2 cos 20 t`
c) At the maximum compressed position, the body is at the extreme left position. The initial phase is `(3pi)/2`
`:. x = a sin (omegat + (3pi)/2) = - a cosomegat = - 2 cos 20t`
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 ______.
If the metal bob of a simple pendulum is replaced by a wooden bob of the same size, then its time period will.....................
- increase
- remain same
- decrease
- first increase and then decrease.
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`
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.
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)
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 mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. [Hint: Start with the equation x = acos (ωt+θ) and note that the initial velocity is negative.]
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 ______.
If the maximum velocity and acceleration of a particle executing SHM are equal in magnitude, the time period will be ______.
The relation between acceleration and displacement of four particles are given below: Which one of the particles is executing simple harmonic motion?
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
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?
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 ______.
