A car of mass $m$ moving over a concave bridge of radius $r$. Find the normal reaction acting on the car when it is at the lowest point of the bridge.

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?