A particle of mass \(m\) is moving along a trajectory given by
\(x = x_0 + a\, cos\,\omega_1 t\)
\(y = y_0 + b\, sin\,\omega_2t\)
The torque, acing on the particle about the origin, at \(t = 0\) is

Muon \((\mu ^{-1})\) is negatively charged \(\left( {\left| q \right| = \left| e \right|} \right)\) with a mass \(m_{\mu}\, = 200\,m_e\), where \(m_e\) is the mass of the electron and \(e\) is the electronic charge. If \(\mu ^{-1}\) is bound to a proton to form a hydrogen like atom, identify the correct statements \((A)\) Radius of the muonic orbit is \(200\) times smaller than that of the electron \((B)\) the speed of the \(\mu ^{-1}\) in the \(n^{th}\) orbit is \(\frac{1}{{200}}\) times that of the election in the \(n^{th}\) orbit \((C)\) The Ionization energy of muonic atom is \(200\) times more than that of an hydrogen atom \((D)\) The momentum of the muon in the \(n^{th}\) orbit is \(200\) times more than that of the electron

Escape velocity of a body from earth is \(11.2 \mathrm{~km} / \mathrm{s}\). If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the plate is _______.

A particle initially at rest starts moving from reference point. \(\mathrm{x}=0\) along \(\mathrm{x}\)-axis, with velocity \(v\) that varies as \(v=4 \sqrt{\mathrm{x} m} / \mathrm{s}\). The acceleration of the particle is _______ \( \mathrm{ms}^{-2}\).

The potential (in volts ) of a charge distribution is given by \(V(z)\, = \,30 - 5{z^2}for\,\left| z \right| \le 1\,m\) \(V(z)\, = \,35 - 10\,\left| z \right|for\,\left| z \right| \ge 1\,m\) \(V(z)\)  does not depend on \(x\)  and  \(y.\)  If this potential is generated by a constant charge per unit volume \(\rho _0\) (in units of \(\varepsilon _0\) ) which is spread over a certain region, then choose the correct statement

Determine the charge on the capacitor in the following circuit ................ \(\mu  C\)

A transverse wave travels on a taut steel wire with a velocity of \({v}\) when tension in it is \(2.06 \times 10^{4} \;\mathrm{N} .\) When the tension is changed to \(T\). the velocity changed to \(\frac v2\). The value of \(\mathrm{T}\) is close to

A force of \(- P \hat{k}\) acts on the origin of the coordinate system. The torque about the point \((2,-3)\) is \(P(a \hat{i}+b \hat{j})\), the ratio of \(\frac{a}{b}\) is \(\frac{x}{2}\). The value of \(x\) is

A parallel plate capacitor with area \(200\,cm^2\) and separation between the plates \(1.5\,cm\), is connected across a battery of \(emf\) \(V\). If the force of attraction between the plates is \(25\times10^{-6}\,N\), the value of \(V\) is approximately........\(V\) \(\left( {{\varepsilon _0} = 8.85 \times {{10}^{ - 12}}\,\frac{{{C^2}}}{{N{m^2}}}} \right)\)

The magnetic flux through a coil perpendicular to its plane is varying according to the relation \(\phi=\) \(\left(5 t^{3}+4 t^2+2 t-5\right) \; Weber\). If the resistant of the coil is \(5 \; ohm\), then the induced current through the coil at \(t =2 \; sec\) will be \(....\,A\)

A hollow spherical shell at outer radius \(R\) floats just submerged under the water surface. The inner radius of the shell is \(r\). If the specific gravity of the shell material is \(\frac{27}{8}\) \(w.r.t.\) water, the value of \(r\) is\(......R\)

For the photoelectric effect, the maximum kinetic energy \(\left(E_{\mathrm{k}}\right)\) of the photoelectrons is plotted against the frequency \((v)\) of the incident photons as shown in figure. The slope of the graph gives _______.

The magnetic field at the centre of a current carrying circular loop of radius \(R\) is 16 \(\mu\)T. The magnetic field at a distance \(x = \sqrt{3}R\) on its axis from the centre is ________ \(\mu\)T.