When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is \(\frac{x}{8}\), where \(x=\) _______.

A prism of angle \(75^{\circ}\) and refractive index \(\sqrt{3}\) is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ___________ .
\(\left(\sin 15^{\circ}=0.25\right.\) and \(\left.\sin 25^{\circ}=0.43\right)\)

For a periodic motion represented by the equation \(Y=\sin \omega t+\cos \omega t\) The amplitude of the motion is 

Assuming the earth to be a sphere of uniform mass density, a body weighed \(300 \mathrm{~N}\) on the surface of earth. How much it would weigh at \(R / 4\) depth under surface of earth?

Following statements are given : \((1)\) The average kinetic energy of a gas molecule decreases when the temperature is reduced. \((2)\) The average kinetic energy of a gas molecule increases with increase in pressure at constant temperature. \((3)\) The average kinetic energy of a gas molecule decreases with increases in volume. \((4)\) Pressure of a gas increases with increase in temperature at constant pressure. \((5)\) The volume of gas decreases with increase in temperature. Choose the correct answer from the options given below:

Three balls of masses \(2 \mathrm{~kg}, 4 \mathrm{~kg}\) and \(6 \mathrm{~kg}\) respectively are arranged at centre of the edges of an equilateral triangle of side \(2 \mathrm{~m}\). The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be _______\(\mathrm{kg} \mathrm{m}{ }^2\).

Electric field of plane electromagnetic wave propagating through a non-magnetic medium is given by \({E}=20 \cos \left(2 \times 10^{10} {t}-200 {x}\right) \,{V} / {m} .\) The dielectric constant of the medium is equal to : (Take \(\mu_{{r}}=1\) )

A proton of mass \(m\) collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of \(90^o\) with respect to each other. The mass of unknown particle is

A bob of mass \('m'\) suspended by a thread of length \(l\) undergoes simple harmonic oscillations with time period \({T}\). If the bob is immersed in a liquid that has density \(\frac{1}{4}\) times that of the bob and the length of the thread is increased by \(1 / 3^{\text {rd }}\) of the original length, then the time period of the simple harmonic oscillations will be :-

Moment of inertia of a body about a given axis is \(1.5\, kg\, m^2\) Initially the body is at rest. In order to produce a rotational kinetic energy of \(1200\, J\), the angular acceleration of \(20\, rad/s^2\) must be applied about the axis of rotation for a duration of ......... \(\sec\).