A uniform disc of radius \(R\) and mass \(M\) is free to rotate only about its axis. A string is wrapped over its rim and a body of mass \(m\) is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is

A gas can be taken from \(A\) to \(B\) via two different processes \(ACB\) and \(ADB\). When path \(ACB\) is used \(60\, J\) of heat flows into the system and \(30\, J\) of work is done by the system. If path \(ADB\) is used work down by the system is \(10\, J\). the heat flow into the system in path \(ADB\) is ..... \(J\)

Two charges each of magnitude \(0.01\,C\) and separated by a distance of \(0.4\,mm\) constitute an electric dipole. If the dipole is placed in an uniform electric field ' \(\vec{E}\) ' of \(10\,dyne/C\) making \(30^{\circ}\) angle with \(\overrightarrow{ E }\), the magnitude of torque acting on dipole is :

A composite parallel plate capacitor is made up of two different dielectric materials with different thickness \(\left(t_{1}\right.\) and \(\left.t_{2}\right)\) as shown in figure. The two different dielectric material are separated by a conducting foil \(F\). The voltage of the conducting foil is \(.....V\)

The initial pressure and volume of an ideal gas are \(P_0\) and \(V_0\). The final pressure of the gas when the gas is suddenly compressed to volume \(\frac{ V _0}{4}\) will be (Given \(\gamma=\) ratio of specific heats at constant pressure and at constant volume)

Two masses of \(3.4\) kg and \(2.5\) kg are accelerated from an initial speed of \(5\) m/s and \(12\) m/s, respectively. The distances traversed by the masses in the \(5^\text{th}\) second are \(104\) m and \(129\) m, respectively. The ratio of their momenta after \(10\) s is \(\dfrac{x}{8}\). The value of \(x\) is ________.

A solution containing active cobalt \({}_{27}^{60}Co\) having activity of \(0.8\,\mu Ci\) and decay constant \(\lambda \) is injected in an animal's body. If \(1 \,cm^3\) of blood is drawn from the animal's body after \(10\, hrs\) of injection, the activity found was \(300\, decays\) per minute. What is the volume of blood that is flowing in the body?..........\(litres\) ( \(ICi = 3.7 \times 10^{10}\) decay per second and at \(t = 10\, hrs\) \({e^{ - \lambda t}} = 0.84\) )

Capacitance of an isolated conducting sphere of radius \(R_{1}\) becomes \(n\) times when it is enclosed by a concentric conducting sphere of radius \(R_{2}\) connected to earth. The ratio of their radii \(\left(\frac{ R _{2}}{ R _{1}}\right)\) is:

Two sources of light emit with a power of \(200 \mathrm{~W}\). The ratio of number of photons of visible light emitted by each source having wavelengths \(300\  \mathrm{nm}\) and \(500 \mathrm{~nm}\) respectively, will be _______.

A photograph of a landscape is captured by a drone camera at a height of 18 km . The size of the camera film is \(2 \mathrm{~cm} \times 2 \mathrm{~cm}\) and the area of the landscape photographed is \(400 \mathrm{~km}^2\). The focal length of the lens in the drone camera is :

A thin ring of \(10\, cm\) radius carries a uniformly distributed charge. The ring rotates at a constant angular speed of \(40\,\pi \,rad\,{s^{ - 1}}\) about its axis, perpendicular to its plane. If the magnetic field at its centre is \(3.8 \times {10^{ - 9}}\,T\), then the charge carried by the ring is close to \(\left( {{\mu _0} = 4\pi  \times {{10}^{ - 7}}\,N/{A^2}} \right)\)

In the given arrangement of a doubly inclined plane two blocks of masses \(\mathrm{M}\) and \(\mathrm{m}\) are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is \(0.25\) . The value of \(\mathrm{m}\), for which \(\mathrm{M}=10\) \(\mathrm{kg}\) will move down with an acceleration of \(2 \mathrm{~m} / \mathrm{s}^2\), is _______. (take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) and \(\left.\tan 37^{\circ}=3 / 4\right)\)