Oscillations

A particle of mass m oscillates along x-axis according to equation x = a t. The nature of the graph between momentum and displacement of the particle is

1. Circle
2. Hyperbola
3. Ellipse
4. Straight line passing through origin

Two particles are oscillating along two close parallel straight lines side by side, with the same frequency and amplitudes. They pass each other, moving in opposite directions when their displacement is half of the amplitude. The mean positions of the two particles lie on a straight line perpendicular to the paths of the two particles. The phase difference is

1. 6
2. 0
3. 2 3

A particle of mass m is released from rest and follows a parabolic path as shown. Assuming that the displacement of the mass from the origin is small, which graph correctly depicts the position of the particle as a function of time? [figure]

1. [figure]
2. [figure]
3. [figure]
4. [figure]

Out of the following functions representing motion of a particle which represents SHM (1) y = sin t - cos t (2) y = sin3 t (3) y = 5 cos ( 3 4 - 3 t ) (4) y = 1 + t + 2 t2

1. Only (1)
2. Only (4) does not represent SHM
3. Only (1) and (3)
4. Only (1) and (2)

The displacement of a particle along the x-axis is given by x = a 2 . The motion of the particle corresponds to

1. simple harmonic motion of frequency /
2. simple harmonic motion of frequency 3 /2
3. non simple harmonic motion
4. simple harmonic motion of frequency /2

The period of oscillation of a mass M suspended from a spring of negligible mass is T. If along with it another mass M is also suspended, the period of oscillation will now be

1. T
2. T 2
3. 2T
4. 2T

A simple pendulum performs simple harmonic motion about x = 0 with an amplitude a and time period T. The speed of the pendulum at x = a/2 will be

1. aT
2. 3 2 aT
3. a 3T
4. a 32T

Which one of the following equations of motion represents simple harmonic motion ? where k, k0, k1 and a are all positive.

1. Acceleration = -k(x + a)
2. Acceleration = k(x + a)
3. Acceleration = - 2x
4. Acceleration = -k0x + k1x2

Two simple harmonic motions of angular frequency 100 and 1000 rad s-1 have the same displacement amplitude. The ratio of their maximum acceleration is

1. 1 : 103
2. 1 : 104
3. 1 : 10
4. 1 : 102

A point performs simple harmonic oscillation of period T and the equation of motion is given by x = a ( t + /6). After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

1. T/3
2. T/12
3. T/8
4. T/6

A particle executes simple harmonic oscillation with an amplitude a. The period of oscillation is T. The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is

1. T/8
2. T/12
3. T/2
4. T/4

A mass of 2.0 kg is put on a flat pan attached to a vertical spring fixed on the ground as shown in the figure. The mass of the spring and the pan is negligible. When pressed slightly and released the mass executes a simple harmonic motion. The spring constant is 200 N/m. What should be the minimum amplitude of the motion so that the mass gets detached from the pan (take g = 10 m/s2). [figure]

1. 10.0 cm
2. any value less than 12.0 cm
3. 4.0 cm
4. 8.0 cm

The particle executing simple harmonic motion has a kinetic energy K0 2 t. The maximum values of the potential energy and the total energy are respectively

1. K0/2 and K0
2. K0 and 2K0
3. K0 and K0
4. 0 and 2K0

The phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is

2. 0.707
3. zero
4. 0.5

A rectangular block of mass m and area of cross-section A floats in a liquid of density . If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period T, then

1. T 1 m
2. T
3. T 1 A
4. T 1

The circular motion of a particle with constant speed is

1. periodic but not simple harmonic
2. simple harmonic but not periodic
3. periodic and simple harmonic
4. neither periodic nor simple harmonic

A particle executing simple harmonic motion of amplitude 5 cm has maximum speed of 31.4 cm/s. The frequency of its oscillation is

1. 4 Hz
2. 3 Hz
3. 2 Hz
4. 1 Hz

Two springs of spring constants k1 and k2 are joined in series. The effective spring constant of the combination is given by

1. k1k2
2. (k1 + k2)/2
3. k1 + k2
4. k1k2/(k1 + k2)

Which one of the following statements is true for the speed v and the acceleration a of a particle executing simple harmonic motion?

1. When v is maximum, a is maximum
2. Value of a is zero, whatever may be the value of v.
3. When v is zero, a is zero
4. When v is maximum, a is zero.

The potential energy of a simple harmonic oscillator when the particle is half way to its end point is

1. 23E
2. 18E
3. 14E
4. 12E

A particle of mass m oscillates with simple harmonic motion between points x1 and x2, the equilibrium position being O. Its potential energy is plotted. It will be as given below in the graph

1. [figure]
2. [figure]
3. [figure]
4. [figure]

The time period of mass suspended from a spring is T. If the spring is cut into four equal parts and the same mass is suspended from one of the parts, then the new time period will be

1. T/4
2. T
3. T/2
4. 2T

In case of a forced vibration, the resonance peak becomes very sharp when the

1. damping force is small
2. restoring force is small
3. applied periodic force is small
4. quality factor is small

Displacement between maximum potential energy position and maximum kinetic energy position for a particle executing simple harmonic motion is

1. a/2
2. +a
3. a
4. -1

When an oscillator completes 100 oscillations its amplitude is reduced to 13 of initial value. What will be its amplitude, when it completes 200 oscillations?

1. 18
2. 23
3. 16
4. 19

A mass is suspended separately by two different springs in successive order, then time periods are t1 and t2 respectively. If it is connected by both springs as shown in figure then time period is t0, the correct relation is [figure]

1. t02 = t12 + t22
2. t0-2 = t1-2 + t2-2
3. t0-1 = t1-1 + t2-1
4. t0 = t1 + t2

The total energy of particle performing SHM depends on

1. k, a, m
2. k, a
3. k, a, x
4. k, x

Two masses MA and MB are hung from two strings of length lA and lB respectively. They are executing SHM with frequency relation fA = 2fB, then relation

1. lA = lB4, does not depend on mass
2. lA = 4lB, does not depend on mass
3. lA = 2lB and MA = 2MB
4. lA = lB2 and MA = MB2

The bob of simple pendulum having length l, is displaced from mean position to an angular position with respect to vertical. If it is released, then velocity of bob at equilibrium position

1. 2gl(1 - )
2. 2gl(1 + )
3. 2gl
4. 2gl

The equations of two waves acting in perpendicular directions are given as x = a ( t + ) and y = a ( t + ), where = + 2, the resultant wave represents

1. a parabola
2. a circle
3. an ellipse
4. a straight line