The position vector of a particle of mass $$\mathrm{m}$$ moving with a constant velocity $$\vec{v}$$ is given by $$\vec{r}=x(t) \hat{i}+b \hat{j}$$, where $$\mathrm{b}$$ is a constant. At an instant, $$\vec{r}$$ makes an angle $$\theta$$ with the $$x$$-axis as shown in the figure. The variation of the angular momentum of the particle about the origin with $$\theta$$ will be

A small ball of mass m is suspended from the ceiling of a floor by a string of length $$\mathrm{L}$$. The ball moves along a horizontal circle with constant angular velocity $$\omega$$, as shown in the figure. The torque about the centre (O) of the horizontal circle is

A small sphere of mass m and radius r slides down the smooth surface of a large hemispherical bowl of radius R. If the sphere starts sliding from rest, the total kinetic energy of the sphere at the lowest point $$\mathrm{A}$$ of the bowl will be [given, moment of inertia of sphere $$=\frac{2}{5} \mathrm{mr}^2$$]

A mouse of mass m jumps on the outside edge of a rotating ceiling fan of moment of inertia I and radius R. The fractional loss of angular velocity of the fan as a result is,