A spring whose unstretched length is \(\ell \) has a force constant \(k\). The spring is cut into two pieces of unstretched lengths \(\ell_1\) and \(\ell_2\) where, \(\ell_1 = n\ell_2\) and \(n\) is an integer. The ratio \(k_1/k_2\) of the corresponding force constants, \(k_1\) and \(k_2\) will be

The Young's modulus of steel wire of radius \(r\) and length \(L\) is \(Y\). If the radius \(r\) and length \(L\) of the wire are doubled then the value of \(Y\)

A block of mass \(2\,kg\) moving on a horizontal surface with speed of \(4\,ms ^{-1}\) enters a rough surface ranging from \(x =0.5\,m\) to \(x =1.5\,m\). The retarding force in this range of rough surface is related to distance by \(F =- kx\) where \(k =12\,Nm ^{-1}\). The speed of the block as it just crosses the rough surface will be ........... \(\,ms ^{-1}\)

Match List\(-I\) with List\(-II\).

List\(-I\)	List\(-II\)
\((a)\) \({R}_{{H}}\) (Rydberg constant)	\((i)\) \({kg} {m}^{-1} {s}^{-1}\)
\((b)\) \(h\) (Planck's constant)	\((ii)\) \({kg} {m}^{2} {s}^{-1}\)
\((c)\) \(\mu_{{B}}\) (Magnetic field energy density)	\((iii)\) \({m}^{-1}\)
\((d)\) \(\eta\) (coefficient of viscocity)	\((iv)\) \({kg} {m}^{-1} {s}^{-2}\)

Choose the most appropriate answer from the options given below:

The initial velocity \(v_{i}\) required to project a body vertically upward from the surface of the earth to reach a height of \(10\, R ,\) where \(R\) is the radius of the earth, may be described in terms of escape velocity \(v_{ e }\) such that \(v_{i}=\sqrt{\frac{x}{y}} \times v_{ e } .\) The value of \(x\) will be ...... .

A mixture of \(2\, moles\) of helium gas (atomic mass \(= 4\, u\)), and \(1\, mole\) of argon gas (atomic mass \(= 40\, u\)) is kept at \(300\, K\) in a container. The ratio of their rms speeds \(\left[ {\frac{{{V_{rms}}{\rm{(helium)}}}}{{{V_{rms}}{\rm{(argon)}}}}} \right]\), is close to

A resonance circuit having inductance and resistance \(2 \times 10^{-4} H\) and \(6.28 \Omega\) respectively oscillates at \(10\, MHz\) frequency. The value of quality factor of this resonator is ......... \([\pi=3.14]\)

A transformer consisting of \(300\) turns in the primary and \(150\) turns in the secondary gives output power of \(2.2\, kW\). If the current in the secondary coils is \(10\, A\), then the input voltage and current in the primary coil are

Monochromatic light is incident on a glass prism of angle \(A\). If the refractive index of the material of the prism is \(μ\), a ray, incident at an angle \(\theta\), on the face \(AB\) would get transmitted through the face \(AC\) of the prism provided

A steel rod has a radius of \(20\,mm\) and a length of \(2.0\,m\). A force of \(62.8\,kN\) stretches it along its length. Young's modulus of steel is \(2.0 \times 10^{11}\,N / m ^2\). The longitudinal strain produced in the wire is \(..........\times 10^{-5}\)

The rms speed of oxygen molecule in a vessel at particular temperature is \(\left(1+\frac{5}{x}\right)^{\frac{1}{2}} v\), where \(v\) is the average speed of the molecule. The value of \(x\) will be:(Take \(\pi=\frac{22}{7}\) )

A particle is oscillating on the \(X-\) axis with an amplitude \(2\, cm\) about the point \(x_0 = 10\, cm\) with a frequency \(\omega \). A concave mirror of focal length \(5\,
cm\) is placed at the origin ( see figure) Identify the correct statements \((A)\) The image executes periodic motion \((B)\) The image executes non-periodic motion \((C)\) The turning points of the image are asymmetric w.r.t the image of the point at \(x =10\,cm\) \((D)\) The distance between the turning points of the oscillation of the image is \(\frac{{100}}{{21}}\)

A \(0.07\,{H}\) inductor and a \(12 \,\Omega\) resistor are connected in series to a \(220 \,{V}, 50 \,{Hz}\) ac source. The approximate current in the circuit and the phase angle between current and source voltage are respectively. [Take \(\pi\) as \(\left.\frac{22}{7}\right]\)