A hemispherical bowl of the radius $R$ is rotating about its axis of symmetry which is kept vertical. A small ball kept in the bowl rotates with the bowl without slipping on its surface. If the surface of the bowl is smooth and the angle made by the radius through the ball with the vertical is $\mathrm{\theta }$. Find the angular speed at which the bowl is rotating.

A particle of mass $m$ is moving with a constant speed $v$ in a circular path in a smooth horizontal plane (plane of the paper) by a spring force as shown in the figure. If the natural length of the spring is $l_0$ and the stiffness of the spring is $k$, find the elongation of the spring.

Ball A is attached to one end of a rigid massless rod, while an identical ball B is attached to the centre of the rod, as figure illustrates. Each ball has a mass $m=0.50 \mathrm{kg},$ and the length of each half of the rod is $L=0.50 \mathrm{m}.$ This arrangement is held by the empty end and is whirled around in a horizontal circle at a constant rate, so each ball is in uniform circular motion. Ball A travels at a constant speed of $v_A=4.0 \mathrm{m} {\mathrm{s}}^{-1}.$ Find the tension in each half of the rod.

A bead of mass m slides along a curved wire lying on a horizontal surface as shown in figure.

$\left(\mathrm{i}\right)$ If the bead slides at constant speed along the wire, draw the vectors representing the force exerted by the wire on the bead at points $\mathrm{A}, \mathrm{B}, \mathrm{and} \mathrm{C}$.
$\left(\mathrm{ii}\right)$ Now consider the bead in figure speeds up with constant tangential acceleration as it moves towards the right. Draw the vectors representing the force on the bead at points $\mathrm{A}, \mathrm{B}, \mathrm{and} \mathrm{C}$.

In a rotor, a hollow vertical cylindrical structure rotates about its axis and a person rests against the inner wall. At a particular speed of the rotor, the floor below the person is removed and the person hangs resting against the wall without any floor. If the radius of the rotor is $2 \mathrm{m}$ and the coefficient of static friction between the wall and the person is $0.2$, find the minimum speed at which the floor may be removed. Take $g = 10 \mathrm{m} {\mathrm{s}}^{-2}$.

A roller-coaster car has a mass of $500 \mathrm{kg}$ when fully loaded with passengers.

(a) If the vehicle has a speed of $20.0 \mathrm{m} {\mathrm{s}}^{-1}$ at point $A$, what is the force exerted by the track on the car at this point?

(b) What is the maximum speed the vehicle can have at point $B$ and still remain on the track?

A $60 \mathrm{kg}$ woman is on a large vertical swing of radius $20 \mathrm{m}$. The swing rotates with constant speed.

(a) At what speed would she feel weightless at the top?

(b) At this speed, what is her apparent weight at the bottom?