A mass \('m'\) is supported by a massless string wound around a uniform hollow cylinder of mass \(m\) and radius \(R\). If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

The current flowing through an ac circuit is given by \(I=5 \sin (120 \pi t) A\) How long will the current take to reach the peak value starting from zero?

Time period of a simple pendulum in a stationary lift is ' \(T\) '. If the lift accelerates with \(\frac{ g }{6}\) vertically upwards then the time period will be ..... (where \(g =\) acceleration due to gravity)

A force \(\overrightarrow{\mathrm{F}}=2 \hat{i}+\mathrm{b} \hat{j}+\hat{k}\) is applied on a particle and it undergoes a displacement \(\hat{i}-2 \hat{j}-\hat{k}\). What will be the value of \(b\), if work done on the particle is zero.

A diatomic gas \((\gamma=1.4)\) does \(100 \mathrm{~J}\) of work in an isobaric expansion. The heat given to the gas is _______.

A spherical liquid drop of radius \(R\) acquires the terminal velocity \(v_1\) when falls through a gas of viscosity \(\eta\). Now the drop is broken into \(64\) identical droplets and each droplet acquires terminal velocity \(v_2\) falling through the same gas. The ratio of terminal velocities \(v_1/v_2\) is ________.

A physical quantity \(p\) is described by the relation \(p\, = a^{1/2}\, b^2\, c^3\, d^{-4}\) If the relative errors in the measurement of \(a, b, c\) and \(d\) respectively, are \(2\% , 1\%, 3\%\) and \(5\%\), then the relative error in \(P\) will be ........... \(\%\)

A particle is projected with velocity \(v_{0}\) along \(x-\) axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e., \(ma =-\alpha x ^{2}.\) The distance at which the particle stops:

Two cylindrical vessels of equal cross-sectional area \(16\,cm ^{2}\) contain water upto herghts \(100\,cm\) and \(150\,cm\) respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is \(......J\) [Take density of water \(=10^{3}\,kg / m ^{3}\) and \(g =10\,ms ^{-2}\) ]

For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is \(1\,kg\), the angular frequency is \(\omega_1\). When the mass block is \(2\,kg\) the angular frequency is \(\omega_2\). The ratio \(\omega_2 / \omega_1\) is