An electric motor of mass 40 kg is mounted on four vertical springs each having a spring constant of 4000 Nm^-1. The period with which the motor vibrates vertically is

• 0.314 s

• 3.14 s

• 0.628 s

• 0.157 s

Two simple harmonic motions are represented by ${\mathrm{y}}_1 = 4 \sin (4\mathrm{\pi t} + \mathrm{\pi }/2)$ and ${\mathrm{y}}_2 = 3 \cos \left(4\mathrm{\pi t}\right)$. The resultant amplitude is

• 7

• 1

• 5

• $2 + \sqrt{3}$

An observer is approaching a stationary source with a velocity $\frac{1}{4}$th of the velocity of sound. Then the ratio of the apparent frequency to actual frequency of source is

• 4 : 5

• 5 : 4

• 2 : 3

• 3 : 2

When a wave travels in a medium, the particle displacement is given by the equation $\mathrm{y} = \mathrm{a} \sin 2\mathrm{\pi } \left(\mathrm{bt} - \mathrm{cx}\right)$ where a, b and c are constants. The maximum particle velocity will be twice the wave velocity, if

• $\mathrm{c} = \frac{1}{\mathrm{\pi a}}$

• $\mathrm{c} = \mathrm{\pi a}$

• b = ac

• $\mathrm{b} = \frac{1}{\mathrm{ac}}$

A progressive wave $\mathrm{y} = \mathrm{A} \sin \left(\mathrm{kx} - \mathrm{\omega t}\right)$ is reflected by a rigid wall at x = 0. Then the reflected wave can be represented by

• y = A sin (kx + ωt)

• y = A cos (kx + ωt)

• y = − A sin (kx − ωt)

• y = − A sin (kx + ωt)

A heater of 220 V heats a volume of water in 5 min. The same heater when connected to 110 V heats the same volume of water in (minute)

• 5

• 20

• 10

• 2.5

Which one of the following graphs represents the variation of electric field with distance r from the centre of a charged spherical conductor of radius R?

38.

Two conducting spheres A and B of radius a and b respectively are at the same potential. The ratio of the surface charge densities of A and B is

• $\frac{\mathrm{b}}{\mathrm{a}}$

• $\frac{\mathrm{a}}{\mathrm{b}}$

• $\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}$

• $\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}$

Three charges 2q, -q, -q are located at the vertices of an equilateral triangle. At the circumcentre of the triangle

• the field is zero but potential is non-zero

• potential is zero and the field is infinity

• the field is non-zero but potential is zero

• both the field and potential are non-zero

C,V,U and Q are capacitance, potential difference, energy stored and charge of a parallel plate capacitor respectively. The quantities that increase when a dielectric slab is introduced between the plates without disconnecting the battery are

• V and C

• V and U

• U and Q

• V and Q