The potential energy of a particle of mass $$m$$ at a distance $$r$$ from a fixed point $$O$$ is given by $$V\left( r \right) = k{r^2}/2,$$ where $$k$$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $$R$$ about the point $$O$$. If $$v$$ is the speed of the particle and $$L$$ is the magnitude of its angular momentum about $$O,$$ which of the following statements is (are) true?

Consider a body of mass $$1.0$$ $$kg$$ at rest at the origin at time $$t=0.$$ A force $$\overrightarrow F = \left( {\alpha t \widehat i + \beta \widehat j} \right)$$ is applied on the body, where $$\alpha = 1.0N{s^{ - 1}}$$ and $$\beta = 1.0\,N.$$ The torque acting on the body about the origin at time $$t=1.0s$$ is $$\overrightarrow \tau .$$ Which of the following statements is (are) true?

A uniform capillary tube of inner radius $$r$$ is dipped vertically into a beaker filled with water. The water rises to a height $$h$$ in the capillary tube above the water surface in the beaker. The surface tension of water is $$\sigma .$$ The angle of contact between water and the wall of the capillary tube is $$\theta .$$ Ignore the mass of water in the meniscus. Which of the following statements is (are) true?

Two infinitely long straight wires lie in the $$xy$$-plane along the lines $$x = \pm R.$$ The wire located at $$x = + R$$ carries a constant current $${I_1}$$ and the wire located at $$x=-R$$ carries a constant current $${I_2}.$$ A circular loop of radius $$R$$ is suspended with its center at $$\left( {0,0,\sqrt 3 R} \right)$$ and in a plane parallel to the $$xy$$-plane. This loop carries a constant current $$I$$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $$ + \widehat j$$ direction. which of the following statements regarding the magnetic field $$\overrightarrow B $$ is (are) true?