A bar magnet has a period of oscillation T. If a similar brass piece of the same mass is placed over it, then the number of oscillations it makes in one second is

• $\frac{1}{\sqrt{2}\mathrm{T}}$

• $\frac{\sqrt{2}}{\mathrm{T}}$

• $\frac{1}{\mathrm{T}}$

• $\frac{2}{\mathrm{T}}$

Two oscillating simple pendululs with time periods T and $\frac{5\mathrm{T}}{4}$ are in phase at a given time. They are again in phase after an elapse of time

• 4 T

• 3 T

• 6 T

• 5 T

If the differential equation for a simple harmonic motion is $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dt}}^2} + 2\mathrm{y} = 0$, the time period of the motion is 

• $\mathrm{\pi }\sqrt{2} \mathrm{s}$

• $\frac{\sqrt{2}}{\mathrm{\pi }} \mathrm{s}$

• $\frac{\mathrm{\pi }}{\sqrt{2}} \mathrm{s}$

• $2\mathrm{\pi } \mathrm{s}$

Identify the wrong statement from the following

• If the length of a spring is halved, the time period of each part becomes $\frac{1}{\sqrt{2}}$ times the original

• The effective spring constant K of springs in parallel is given by $\frac{1}{\mathrm{K}} = \frac{1}{{\mathrm{K}}_1} + \frac{1}{{\mathrm{K}}_2} + .......$

• The time period of a stiffer spring is less than that of a soft spring

• The spring constant is inversely proportional to the spring length

The total energy of the particle executing simple harmonic motion of amplitude A is 100 J. At a distance of 0.707 A from the mean position, its kinetic energy is

• 25 J

• 50 J

• 100 J

• 12.5 J

A particle is executing simple harmonic motion with amplitude A. When the ratio of its kinetic energy to the potential energy is $\frac{1}{4}$, its displacement from its mean position is

• $\frac{2}{\sqrt{5}} \mathrm{A}$

• $\frac{\sqrt{3}}{2} \mathrm{A}$

• $\frac{3}{4} \mathrm{A}$

• $\frac{1}{4} \mathrm{A}$

The ratio of amplitudes of two simple harmonic motions represented by the equations ${\mathrm{y}}_1 = 5 \sin \left(2\mathrm{\pi t} + \frac{\mathrm{\pi }}{4}\right)$ and ${\mathrm{y}}_2 = 2\sqrt{2} \left(\sin 2\mathrm{\pi t} + \cos 2\mathrm{\pi t}\right)$ is

• 1 : 1

• 2 : 1

• 5 : 2

• 5 : 4

The displacement of a particle in SHM is $\mathrm{x} = 10 \sin \left(2\mathrm{t} - \frac{\mathrm{\pi }}{6}\right)$ m. When its displacement is 6 m, the velocity of the particle (in ms^-1) is

• 8

• 24

• 16

• 10

Two pendulums of lengths 1.44 m and 1 m start to swing together. The number of vibrations after which they will again start swinging together is

• 4

• 3

• 5

• 2

The average total energy in one time period of a particle of mass m executing SHM of amplitude a and angular velocity ω is

• $\frac{1}{2} {\mathrm{m\omega }}^2{\mathrm{a}}^2$

• $\frac{1}{4} {\mathrm{m\omega }}^2{\mathrm{a}}^2$

• 0

• mω²a²