A uniform rod of mass 250 g having length 100 cm is balanced on a sharp edge at 40 cm mark. A mass of 400 g is suspended at 10 cm mark. To maintain the balance of the rod, the mass to be suspended at 90 cm mark, is

A bottle has an opening of radius \(a\) and length \(b\). A cork of length band radius \(\left( {a + \Delta a} \right)\) where \(\left( {\Delta a <  < a} \right)\) is compressed to fit into the opening completely (see figure). If the bulk modulus of cork is \(B\) and frictional coefficient between the bottle and cork is \(\mu \) then the force needed to push the cork into the bottle is

The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\)

A rectangular loop of sides \(12 \mathrm{~cm}\) and \(5 \mathrm{~cm}\), with its sides parallel to the \(x\)-axis and \(y\)-axis respectively moves with a velocity of \(5 \mathrm{~cm} / \mathrm{s}\) in the positive \(\mathrm{x}\) axis direction, in a space containing a variable magnetic field in the positive \(z\) direction. The field has a gradient of \(10^{-3} \mathrm{~T} / \mathrm{cm}\) along the negative \(\mathrm{x}\) direction and it is decreasing with time at the rate of \(10^{-5} \mathrm{~T} / \mathrm{s}\). If the resistance of the loop is \(6 \mathrm{~m} \Omega\), the power dissipated by the loop as heat is _______ \(\times 10^{-9} \mathrm{~W}\).

A meter bridge with two resistances \( R_{1} \) and \( R_{2} \) as shown in figure was balanced (null point) at 40 cm from the point P. The null point changed to 50 cm from the point P, when 16 \( \Omega \) resistance is connected in parallel to \( R_{2} \). The values of resistances \( R_{1} \) and \( R_{2} \) are __________ .

The maximum error in the measurement of resistance, current and time for which current flows in an electrical circuit are \(1 \%, 2 \%\) and \(3 \%\) respectively. The maximum percentage error in the detection of the dissipated heat will be

A massless spring \((k = 800\, N/m)\), attached with a mass \((500\, g)\) is completely immersed in \(1\, kg\) of water. The spring is stretched by \(2\, cm\) and released so that it starts vibrating. What would be the order of magnitude of the change in the temperature of water when the vibrations stop completely? (Assume that the water container and spring receive negligible heat and specific heat of mass \(= 400\, J/kg\, K\), specific heat of water \(= 4184\, J/kg\, K\))

A simple pendulum made of a bob of mass \(m\) and a metallic wire of negligible mass has time period \(2s\) at  \(T = 0\,^oC\) . If the temeprature of the wire is increased and the corresponding change in its time peirod is plotted against its temperature, the resulting graph is a line of slope \(S\). If the coefficient of linear expansion of metal is \(\alpha \) then the value of  \(S\) is

Two cells of emfs \(1\) V and \(2\) V and internal resistance \(2\,\Omega\) and \(1\,\Omega\), respectively connected in parallel, gave a current of \(1\) A through an external resistance. If the polarity of one cell is reversed, then value of current through the external resistance will be \(\dfrac{\alpha}{5}\) A. The value of \(\alpha\) is __________.

\(64\) identical drops each charged upto potential of \(10\,mV\) are combined to form a bigger dorp. The potential of the bigger drop will be \(..........\,mV\)