Consider a frame that is made up of two thin massless rods $AB$ and $AC$ as shown in the figure. $A$ vertical force $\overrightarrow{ P }$ of magnitude $100 \;N$ is applied at point $A$ of the frame. Suppose the force is $\overrightarrow{ P }$ resolved parallel to the arms $AB$ and $AC$ of the frame. The magnitude of the resolved component along the arm $AC$ is $xN$. The value of $x$, to the nearest integer, is …………

[Given : $\sin \left(35^{\circ}\right)=0.573, \cos \left(35^{\circ}\right)=0.819$ $\left.\sin \left(110^{\circ}\right)=0.939, \cos \left(110^{\circ}\right)=-0.342\right]$

Adjoining figure shows a force of $40\, N$ acting at $30^o$ to the horizontal on a body of mass $5 \,kg$ resting on a smooth horizontal surface. Assuming that the acceleration of free-fall is $10\, ms^{-2}$, which of the following statements is (are) correct?

$[1]$ The horizontal force acting on the body is $20\, N$

$[2]$ The weight of the $5\, kg$ mass acts vertically downwards

$[3]$ The net vertical force acting on the body is $30\, N$

As shown in figure, a $70\,kg$ garden roller is pushed with a force of $\overrightarrow{ F }=200\,N$ at an angle of $30^{\circ}$ with horizontal. The normal reaction on the roller is $…….\,N$ $\left(\right.$ Given $\left.g =10\,m s ^{-2}\right)$

An object is subjected to a force in the north-east direction. To balance this force, a second force should be applied in the direction

Give the magnitude and direction of the net force acting on

$(a)$ a drop of rain falling down with a constant speed,

$(b)$ a cork of mass $10\; g$ floating on water,

$(c)$ a kite skillfully held stationary in the sky,

$(d)$ a car moving with a constant velocity of $30\; km/h$ on a rough road,

$(e)$ a high-speed electron in space far from all material objects, and free of electric and magnetic fields.