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

33.

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

Two travelling waves, y₁ = A sin [k(x + ct)] and y₂ = A sin [k(x- ct)] are superposed on a string. The distance between adjacent antinodes is

• $\frac{\mathrm{ct}}{\mathrm{\pi }}$

• $\frac{\mathrm{ct}}{2\mathrm{\pi }}$

• $\frac{\mathrm{\pi }}{\mathrm{k}}$

• $\frac{\mathrm{k}}{\mathrm{\pi }}$

If a stretched wire is vibrating in the second overtone, then the number of nodes and antinodes between the ends of the string are respectively

• 2 and 2

• 1 and 2

• 3 and 4

• 2 and 3

Pick out the correct statement in the following with reference to stationary wave pattern

• In a tube closed at one end, all the harmonics are present

• In a tube open at one end, only even harmonics are present

• The distance between successive nodes is equal to the wavelength

• In a stretched string, the first overtone is the same as the second harmonic

Five cells each of emf E and internal resistance r send the same amount of current through an external resistance R whether the cells are connected in parallel or in series. Then the ratio $\left(\frac{\mathrm{R}}{\mathrm{r}}\right)$ is

• 2

• $\frac{1}{2}$

• $\frac{1}{5}$

• 1

The power dissipated in the transmission cables carrying current I and voltage V is inversely proportional to

• V

• V²

• $\sqrt{\mathrm{V}}$

• $\sqrt{\mathrm{I}}$

The magnitude of the magnetic field inside a long solenoid is increased by

• decreasing its radius

• decreasing the current through it

• increasing its area of cross-section

• introducing a medium of higher permeability

A bar magnet of moment of inertia 9 x 10^-5 kg m² placed in a vibration magnetometer and oscillating in a uniform magnetic field 16π² × 10^-5 T makes 20 oscillations in 15 s. The magnetic moment of the bar magnet is

• 3 A-m²

• 2 A-m²

• 4 A-m²

• 6 A-m²