A particle is moving in a circular path of radius \(a,\) with a constant velocity \(v\) as shown in the figure.The centre of circle is marked by \('C'\). The angular momentum from the origin \(O\) can be written as

The diameter and height of a cylinder are measured by a meter scale to be \(12.6 \pm 0.1\, cm\) and \(34.2  \pm  0.1\, cm\), respectively. What will be the value of its volume in appropriate significant figures?

Identify the correct statements from the following: \((A)\) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative. \((B)\) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative. \((C)\) Work done by friction on a body sliding down an inclined plane is positive. \((D)\) Work done by an applied force on a body moving on a rough horizontal plane with uniform velocity in zero. \((E)\) Work done by the air resistance on an oscillating pendulum in negative. Choose the correct answer from the options given below:

A particle is moving in a circular path of radius a under the action of an attractive potential \(U = - \frac{k}{{2{r^2}}}\) Its total energy is

At an angle of \(30^{\circ}\) to the magnetic meridian, the apparent dip is \(45^{\circ} .\) Find the true dip :

A helicopter flying horizontally with a speed of \(360 \mathrm{~km} / \mathrm{h}\) at an altitude of 2 km , drops an object at an instant. The object hits the ground at a point O , 20 s after it is dropped. Displacement of ' O ' from the position of helicopter where the object was released is :
(use acceleration due to gravity \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) and neglect air resistance)

A \(d.c.\) main supply of \(e.m.f.\, 220\, V\) is connected across a storage battery of \(e.m.f.\, 200\, V\) through a resistance of \(1\,\Omega \). The battery terminals are connected to an external resistance \('R'\) . The minimum value of \('R'\), so that a current passes through the battery to charge it is ............... \(\Omega\)

The alternating current is given by \({i}=\left\{\sqrt{42} \sin \left(\frac{2 \pi}{{T}} {t}\right)+10\right\} {A}\) The \(r.m.s.\) value of this current is \({A}\)

If \(a _{1}, a _{2}, a _{3} \ldots\) and \(b _{1}, b _{2}, b _{3} \ldots\) are \(A.P.\) and \(a_{1}=2, a_{10}=3, a_{1} b_{1}=1=a_{10} b_{10}\) then \(a_{4} b_{4}\) is equal to

Young's moduli of the material of wires \(A\) and \(B\) are in the ratio of \(1: 4\), while its area of cross sections are in the ratio of \(1: 3\). If the same amount of load is applied to both the wires, the amount of elongation produced in the wires \(A\) and \(B\) will be in the ratio of [Assume length of wires \(A\) and \(B\) are same]

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to

Match the LIST-I with LIST-II

LIST-I	LIST-II
(A) Gravitational constant	(I) \(\left[\mathrm{LT}^{-2}\right]\)
(B) Gravitational potential energy	(II) \(\left[\mathrm{L}^2 \mathrm{~T}^{-2}\right]\)
(C) Gravitational potential	(III) \(\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]\)
(D) Acceleration due to gravity	(IV) \(\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]\)

Choose the correct answer from the options given below :

The electric field in a plane electromagnetic wave is given by \(\overrightarrow{{E}}=200 \cos \left[\left(\frac{0.5 \times 10^{3}}{{m}}\right) {x}-\left(1.5 \times 10^{11} \frac{{rad}}{{s}} \times {t}\right)\right] \frac{{V}}{{m}} \hat{{j}}\) If this wave falls normally on a perfectly reflecting surface having an area of \(100 \;{cm}^{2}\). If the radiation pressure exerted by the \(E.M.\) wave on the surface during a \(10\, minute\) exposure is \(\frac{{x}}{10^{9}} \frac{{N}}{{m}^{2}}\). Find the value of \({x}\).