An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at \(30^{\circ} C\). The coefficient of linear expansion of aluminium and steel are \(24 \times 10^{-6} /{ }^{\circ} C\) and \(1.2 \times 10^{-5} /{ }^{\circ} C\), respectively. The length of this composite rod when its temperature is raised to \(100^{\circ} C\), is _________ cm .

When a part of a straight capillary tube is placed vertically in a liquid, the liquid raises upto certain height h. If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by 1% each, then the height of the liquid in the tube will change by __________ %.

Two discs of moment of inertia \(I_1=4 \mathrm{~kg} \mathrm{~m}^2\) and \(\mathrm{I}_2=2 \mathrm{~kg} \mathrm{~m}^2\) about their central axes & normal to their planes, rotating with angular speeds \(10 \mathrm\ {rad} / \mathrm{s}\) & \(4  \mathrm\ {rad} / \mathrm{s}\) respectively are brought into contact face to face with their axe of rotation coincident. The loss in kinetic energy of the system in the process is _______ \(\mathrm{J}\).

A point object, \(O\) is placed in front of two thin symmetrical coaxial convex lenses \(L _1\) and \(L _2\) with focal length \(24\,cm\) and \(9\,cm\) respectively. The distance between two lenses is \(10\,cm\) and the object is placed \(6\,cm\) away from lens \(L _1\) as shown in the figure. The distance between the object and the image formed by the system of two lenses is .........\(cm\)

A boy ties a stone of mass \(100 \,g\) to the end of a \(2\) \(m\) long string and whirls it around in a horizontal plane. The string can withstand the maximum tension of \(80 \,N\). If the maximum speed with which the stone can revolve is \(\frac{ K }{\pi} rev\). / \(min\). The value of \(K\) is (Assume the string is massless and unstretchable)

Two metal plates \((A, B)\) are kept horizontally with separation of \(\left(\dfrac{12}{\pi}\right)\) cm, with plate A on the top. An atomizer jet sprays oil (density \(1.5\) g/cm\(^3\)) droplets of radius \(1\) mm horizontally. All oil droplets carry a charge \(5\) nC. The potentials \(V_A\) and \(V_B\) are required on plates A and B respectively in order to ensure the droplets do not descend. The values of \(V_A\) and \(V_B\) are _______.
(Neglect the air resistance to the droplets and take \(g = 10\) m/s\(^2\))

Consider a circular loop that is uniformly charged and has a radius \(\mathrm{a} \sqrt{2}\). Find the position along the positive z -axis of the cartesian coordinate system where the electric field is maximum if the ring was assumed to be placed in xy-plane at the origin :

The speed verses time graph for a particle is shown in the figure. The distance travelled (in \(m\) ) by the particle during the time interval \(t=0\) to \(t=5\, s\) will be

The following observations were taken for determining surface tension \(T\) of water by capillary method: diameter of capillary, \(D= 1.25 \times 10^{-2}\; m\) rise of water, \(h=1.45 \times 10^{-2}\; m \)  Using \(g= 9.80 \;m/s^2\) and the simplified relation \(T = \frac{{rhg}}{2}\times 10^3 N/m\) , the possible error in surface tension is ........... \(\%\)

A uniform rod of mass m and length \(l\) suspended by means of two identical inextensible light strings as shown in figure. Tension in one string immediately after the other string is cut, is _________.(g acceleration due to gravity)

A thin uniform tube is bent into a circle of radius \(r\) in the virtical plane. Equal volumes of two immiscible liquids, whose densities are \({\rho _1}\) and \({\rho _2}\left( {{\rho _1} > {\rho _2}} \right)\) fill half the circle. The angle \(\theta\) between the radius vector passing through the common interface and the vertical is